Downloading it should only save it to somewhere you can access the file, but installing it might be a different story. Some programs will install globally (for all users) and some will install only for you, (one user).
Regrettably there's not many ways you can predict how the installer will act unless it asks you "Install for everyone or for just one user?"
I avoid letting programs put icons on the desktop, because if it's a global program, it'll put the icon in the shared desktop folder. (requiring administrator privileges to remove in vista and windows 7). As far as I know, it's safe to let it put icons in the quick launch bar, if you use that and on the start menu itself. This way you can find it on your machine but anyone else who might use the machine probably won't notice the new program.
Programs don't take up much space, and if your hard disk has quotas set to prevent overusage by each user, then don't worry, programs install as the user "TrustedInstaller" most of the time, so it's installation probably won't count against your hard disk limits.