it depends on the game, Squad. Quake 3, for example, had a variety of mods, some of which included bots for off-line play, whilst others were dependent on online play.
games like Q3, Q2, Unreal Tournament, UT2K4, Elite Force and Jedi Knight 2 all had downloadable characters, with the UT games you were more dependent on them being on the servers before you could use them. with the Quake series and the various Q3 engine games - EF1, JK2, RTCW - you could use 'skins' and models without needing them to be on the server, but if no-one had that skin or model or had downloads turned on, they'd see you as a default character.
i used to be really into all that skin/model stuff and was a long-time member on Polycount.com where they had tons of Q2, Q3 and UT1 stuff, but their hosts, Gamespy, got hacked and the site was wrecked. it's never been the same since...
i helped do bot support for some Q3 models including Sonic the Hedgehog, Mario and a load of original characters. i also did a load for JK2 skins and models.
your best bet is to check out the likes of q3files.com effiles.com or ut2004files.com for stuff.
there's a guy called Laz Rojas who's done a shitload of quality EF1 stuff, including the TNG and TOS casts, and LOADS more including Bajorans, TOS Klingons, TOS Romulans, Khan, Andorians, Vulcans and Ferengi. he even did the Nomad probe from 'The Changeling' in a TOS villains pack!
http://elitefarce.planets.gamespy.com/ - his site.