I like both ideas above. Maybe have everyone over and record messages to the new parents and instead of gifts that have to be shipped, everyone brings an envelope with a monetary gift and you can in turn get a prepaid visa card with the money to send with the video. You wouldn't have to reveal who gave what amount. They don't even have to put their names on it. You just open them when everyone leaves. They would know who contributed simply by the video messages.
As far as money for having people over, make it a Potluck Shower! I don't think that would be rude, not sure. To get the word out about the gathering, look online for free e-vites.