Ah, now here's something I can speak to.
My collection includes very few prop replicas, and even fewer fan-made ones. It's mostly toys. However, after joking about the "museum" in my home for some years, a few friends of mine and I got together and made it official: we formed the non-profit Star Toys Museum, Inc. to preserve, protect, and share the collection.
It's still in my home, still very disorganized and in need of much more room (there have been three new Star Wars movies since we made The Museum official, four if you count The Clone Wars), but we're here and still receive visitors. In fact, the last month or so has seen a veritable explosion in visitors -- I don't know why, but I suspect the economy has folks looking for inexpensive things to do close to home.
We don't have set hours, and don't charge admission. We're open by appointment -- send an email or leave a phone message, and we can usually arrange for someone to greet visitors -- and we have a jar so folks can leave contributions, if they wish.
Two years ago, the Geppi's Entertainment Museum in downtown Baltimore borrowed a boatload of our toys for their Star Wars 30th Anniversary exhibit. They were very professional, insuring and cataloguing everything. It was a blast to see my toys in their facility (my avatar picture is from opening night there). Since then, I've been slowly putting together a "traveling collection", so that the next time someone wants to do something like that, we won't have to take apart any of our displays -- we'll just have boxes of toys ready to go.
Here's a link:
http://www.startoysmuseum.org/
I know that the folks at Geppi's, at the Kruger Street Toy and Train Museum of Wheeling, WV, and the Antique Toy Museum of Stanton, MO are living my dream, and I'm sure there are others as well.