This list probably is not complete, but this is everything I know of right now.  Please tell me any that are missing! Thanks.
Edited In Additions
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12/10/2014: Horror Story (Turbo CD) (J only) – This is an autoscrolling side-scrolling run & gun platform-action game. It has two player co-op.
12/12/2014: Cosmo Gang: The Video (SNES) (J only) – Galaga-style static-screen shmup from Namco with two player co-op.
This list is broken up into two parts. First, I list all of the shmups, sorted by platform and release region. Second, below the shmup list, I list all of the arena shooters, run & guns, rail shooters, and freeroam 3d shooters, again sorted by platform and release region.
Key
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U means North America. E means Europe or PAL-land (so it usually includes Australia). J means Japan.
[Games with only alternating and not simultaneous multiplayer titles are marked with brackets] while the games with better multiplayer, that is simultaneous, do not have those brackets.
Totals
These will be listed from most to least, with games for addons included in each main system’s total. The Turbografx plus Turbo CD and SuperGrafx has the most: 26 games on the list below with simultaneous two-player multiplayer, combining both shmups and the other genres, plus 4 games with alternating multiplayer (Only one is a SuperGrafx title, if that platform was to be listed separately; it would not change the order of this list.). Second, the NES plus FDS has 16 with simultaneous multiplayer (one of which has 3 player multiplayer in its Japanese release), plus 9 with alternating-only multiplayer. Tied with it, the SNES has a total of 19 games with simultaneous and 6 games with alternating multiplayer. Following it, the Genesis plus Sega CD and 32X has 17 games with simultaneous multiplayer. Farther down the list, the Sega Master System has 6 with simultaneous multiplayer, plus 2 with alternating only. Next the Atari 7800 has 6, including one unreleased but available title.  Last, the Colecovision, Vectrex, Atari 5200, Sega SG-1000, and Phillips CD-i do not have any I know of, though there likely are some I am missing from this list.
List: 3rd and 4th Gen Multiplayer Shmups
The early 3rd gen (sometimes wrongly called “2nd” or “2.5” gen) consoles of 1982-1983 that aren’t the NES — that is, the Atari 5200, Colecovision, Vectrex, and Sega SG-1000, to name the major ones — do these systems have any at all? I can’t find any so far.
NES
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U
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Silkworm
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JU/JUE
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Stinger (3 player in Japan, 2 player in US)
Life Force (UE) / Salamander (J)
S.C.A.T. (J title: Final Mission; E title: Action in New York)
Legendary Wings
[Seicross]
[Gradius]
[Image Fight]
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E/JE
—
None
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J
—
Twinbee
Twinbee 3
Crisis Force
[Gradius II: Gofer no Yabou]
[Insector X]
[Zunou Senkan Galg]
Atari 7800
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Centipede (static screen)
Asteroids (static screen/arena)
Plutos (unreleased, but the rom leaked years ago)
Sega Master System
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UE
—
[Cloud Master]
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E
—
Super Space Invaders
[Scramble Spirits]
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J
—
Satellite 7
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Korea Only
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Cyborg Z
SNES
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JU/JUE
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Firepower 2000 (Super SWIV)
Raiden Trad
Aero Fighters (JP title: Sonic Wings)
Darius Twin
Strike Gunner S.T.G.
Space Invaders: The Original Game (static screen) (multiplayer is competitive versus play, not co-op)
[Gradius III]
[R-Type III: The Third Lightning]
—
JE
—
Pop’n Twinbee
[Parodius: Non-Sense Fantasy]
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J
—
Cosmo Gang: The Video (static screen)
[Gokujou Parodius]
[Jikkyou Oshaberi Parodius]
Turbografx-16
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JU
—
Ordyne
Aero Blasters
[Psychosis]
[Fantasy Zone]
—
J
—
Salamander
Detana! Twinbee
Dragon Saber – The After Story of Dragon Spirit
1943 Kai
Burning Angels
Toilet Kids
Toy Shop Boys
[Gradius]
SuperGrafx
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J
—
1941: Counter Attack
Turbo CD
Note: CD: Regular CD title. SCD: Super CD title. ACD: Arcade CD title.
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JU
—
Buster Bros. (static shooter) (CD)
Forgotten Worlds (p2 plays as the option) (SCD)
Monster Lair (part shmup, part autoscrolling platform-shooter) (CD)
—
J
—
Hellfire S: The Another Story (CD)
Ginga Fukei Densetsu Sapphire (ACD)
Legion (CD)
Space Invaders: The Original Game (SCD) (static screen; multiplayer is competitive versus play, not co-op)
Psychic Storm (SCD)
Ai Cho Aniki (SCD)
Metamor Jupiter (SCD) (code unlocks a 3 player mode — others play as the options)
[Gradius II: Gofer no Yabou (SCD)]
Genesis
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U/UE
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Battle Squadron
Gadget Twins (U only)
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JU/JUE
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Bio Hazard Battle
Air Buster (JP title: Aero Blasters)
Forgotten Worlds
Vapor Trail (JP title: Kuuga: Operation Code – “Vapor Trail”
—
E/JE
—
Mega SWIV (E only)
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J
—
Bio-Ship Paladin
Sega CD
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None
32X (JUE)
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Zaxxon’s Motherbase 2000 (multiplayer is competitive versus play, not co-op)
CD-i
—
None
Multiplayer [side-scrolling] Run & Guns, topdown run & gun-like action games, rail shooters, and freeroam 3d shooters on 3rd and 4th gen consoles
{each games’ genre is in these brackets}
[square brackets still mean alternating multiplayer]
NES
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U
—
Contra Force {run & gun}
—
JUE
—
Contra {run & gun}
Super C {run & gun}
Ikari Warriors {topdown}
Ikari Warriors 2: Victory Road {topdown}
Ikari Warriors 3 {topdown / beat ’em up}
Guerilla War (JP: Guevara) {topdown}
Smash T.V. {arena}
[Front Line] {topdown}
[Seicross] {isometric}
Famicom Disk System (J only)
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[Kiki Kaikai: Doutou-hen] {topdown}
SMS
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UE
—
Rambo: First Blood Part II (J title: Ashura; E title: Secret Command) {topdown}
Time Soldiers {topdown}
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E
—
Super Smash T.V. {arena}
7800
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Commando {topdown}
Ikari Warriors {topdown}
Robotron 2084 {arena}
SNES
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JU/JUE
—
Pocky & Rocky {topdown}
Pocky & Rocky 2 {topdown}
Sunset Riders {run & gun}
Contra 3: The Alien Wars {run & gun}
Soldiers of Fortune (E title: The Chaos Engine) {topdown}
Wild Guns {run & gun/lightgun}
GunForce {run & gun}
Nightmare Busters (homebrew release) {run & gun}
Super Smash T.V. {arena}
Total Carnage {topdown/arena}
—
J
—
[Galaxy Wars] {weird static-screen shmuplike where you control each missile you’re trying to hit the enemy ships with}
Deae Tonosama Appare Ichiban {topdown}
Genesis
—
UE
—
The Adventures of Batman & Robin {run & gun}
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JUE
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Contra: Hard Corps {run & gun}
Gunstar Heroes {run & gun}
Rolling Thunder 2 {sidescrolling shooter}
Soldiers of Fortune (E title: The Chaos Engine) {topdown}
Sunset Riders (different from the SNES game) {run & gun}
Super Smash T.V. {arena}
Sega CD (JUE)
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SoulStar (P2 just controls a cursor, lightgun style) {rail/freeroam flight combat.}
32X (JUE)
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Star Wars Arcade (P2 just controls a cursor, lightgun style) {rail/freeroam flight combat}
TurboGrafx-16
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JU
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Shockman (JP: Kaizou Ningen Shubibinman 2) {run & gun / sidescrolling action-platformer}
—
J
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Kaizou Ningen Shubibinman {run & gun / sidescrolling action-platformer}
Turbo CD
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J
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Mystic Formula {topdown}
Galaxy Keiji Gayvan {run & gun}
Kaizou Ningen Shubibinman 3: Ikai no Princess {run & gun / sidescrolling action-platformer}
Horror Story {autoscrolling run & gun action-platformer}
Hey Brian
Kollision from Sega-16 here.
This is a very nice list! 🙂
I never heard of this Cyborg Z for the Master System, now I’ll have to keep an eye out for it.