I'm looking for an arcade racing game to play on my home LAN, 1NSANE seems to be an ideal choice.
Will I need to buy more than one copy to play on the LAN only (not internet play) ?

I shouldn't think so. That's the beauty of DRM Free! :)

Well, you don't *have* to, but that would not be nice to GOG and the publishers/developers. To be fair and legal(?), yes, each person should own a copy of the game. Otherwise, the system here will get abused too often.
But since people are going to do this anyway: At the very least, let everyone on your LAN know of this site and where they can get their own copy. Who knows, you may find the majority will want some other game by looking at the catalogue.
**warning** Some games come with Unique keys, and may not work if you do this.

Well, if I remember right, on one of the pages GOG states that you are allowed to put the games on any/all computers within your home (not business though, or your buddy's laptop he brought over). So as long as all computers involved are in your own house, apparently you're ethically clear to run on one copy, apart from any CD-Key issues Ois mentioned. Although if you want to, I'm sure GOG and the publishers/developers will be happy to sell you more copies. :)
Edit: Link to the GOG page is this:
http://www.gog.com/en/support/website_help/downloads_and_games

My eyes focused on this first then I had a really dirty thought. I made need mental help.

It has been mentioned allready, everybody on the LAN can use the same install file and you're all able to play it in LAN (if theres no unique key).
I have the same problem, I realy would like to play UT 2004 on the next lan but I'm the only person who has a working copy from gog. But for me it doesn't feel right to spread the UT 2004 installer around, I think everybody who wants to play a game should buy his own copy.
I thought about giving my friends the installer but to delete the installer and games after 1-2 hours UT. If they want more UT they have to buy it by themself. I think it's just fair to gog and the developers. Besides it's very important for the future of gog. If most people start giving their games to their friends other publishers might not want to publish their games on gog and we will miss many other games. So I try to support gog as much as I can. I have a dream, that one day I can find all classic games from the 90s and the early 200x here :).
If you're looking for an arcade racer, you should take a look at Flatout (link to the demo http://www.gamershell.com/download_7433.shtml ). With its mini games its great on LANs and its just $ 5,99. So it will be easier to convince your friends to buy it :).

Reading through the link Luned posted, and the OP again.
It looks okay to run multiple copies on any of your own non-business based machines. If you own all the PCs on the home LAN, and don't give out the installer you should be fine.
If a friend comes around with their own PC, then they will need their own purchased copy.
This is actually pretty nice, but when GOG staff return it would be good to hear what they say. Looks good though.

Everyone who wants to play ought to pay for their own copy, especially if it installed on their own machine. (Meaning they end up with it free after the party itself.)

Tell your friends to buy the game instead of cigarettes, or pizza, or something else worth 5.99 or 9.99 bucks....it is so cheap that they should be able to afford it.