so will this slow down a shared server too much?
If I get it right, you're not going to run the games on the server, you'll just offer offer them for download. If so, bandwidth is indeed your main obvious requirement. In a shared hosting environment though, if you're asking too much from the hard drives, you'll be affecting other customers. Chances are that things will get to a point where the host notices this, quite soon. I imagine that when you release a new game, or batch of games, the number of downloads will spike, which is yet another problem.
IMHO, you could start with shared hosting, but you need to be ready to upgrade. You could also use subdomains and host them on a number of shared hosting accounts, to spread the load. Alternatively, if you want to keep your costs low, there are file hosting services out there, such as rapidshare.com.