oh, man, I spent countless hours looking for such games, to play with my friend. I avoided fighting games, though, I like Contra-style run'n'gun - my type of thing.
our favorite so far has been a game called The Red Star. It's a Contra-clone, but done very good. it has some RPG aspects, it has a couple of interesting fights. I have to warn you though, it's very difficult. sometimes it seems unfair. it's not a very well designed game, but it's certainly entertaining.
next stop, Champions of Norrath. it supports up to 4 players. and it has an online mode. it's a spin-off in the Everquest universe. think Diablo in 3D, very simple game but enjoyable non the less. there are actually two games in teh series, the other one is called Return to Arms. but they play very much the same. they are very similar to Baldur's gate: Dark Alliance. you might want to check it and it's sequel, Dark Alliance 2.
then there's Hunter: The Reckoning - Wayward. it's somewhat similar to The Red Star, your characters are equipped with both melee and ranged weapons. but personally I didn't like it, it starts off good but gets tedious very fast. and the controls are awful. if you can't rent it, don't bother.
Raging Blades can be fun, a simple anime-style hack'n'slash. it's relatively difficult, so if you're an old-school type of gamer, definitely try it.
Gradius V - a side-scroller set in space, this game is beautiful and kinda hard too. worth checking out, if you enjoy this kind of games.
Gauntlet. there are two games for the PS2 with this title, Seven Sorrows and Dark Legacy. I liked them, reminded me of Golden Age. but I haven't had time to really sit down and play them. the first impression is good, though.
Lord Of The Rings: The Return Of The King. my friend likes it very much. it's one of the better co-op games for the PS2. very well done, not too difficult, has a couple of game modes. interesting missions too. the only downside: it's based on the movie, with respective actors acting as character models.
a lot of people like Obscure (1 and 2). it's a Silent Hill-type of game with a co-op mode. haven't tried it yet myself, though.
that's it for adventure-style games for PS2. only a few left. Urban Reign could work for you. a fighting game. it supports up to 4 players, which is a plus. tons of modes. basically it's like tekken tag tournament but with a different flavor.
need I say anything about Twisted Metal? Head On is fantastic, Black - not so much.
as for PS1, I only have a couple of games. the Metal Slug compilation is a must-have. Fighting Force 1 and 2, both good games if you like stuff like Streets Of Rage and Double Dragon.
my personal favorites are Rogue Trip (a Twisted Metal clone, made by the original Twisted Metal team, it's fantastic) and Loaded (a very intense and stylish game, the best shoot'em-up on PS1).
