Over the weekend I visited a nice little natural history museum. Museums in Libertopia? Of course!
Obviously they are all privately funded. Some are profit motivated while others are supported by a charitable group. Both kinds can be very educational and fun!
Everyone is a lot richer now since there is no "taxation" at any step of the production, and therefore the things you need and want cost a lot less. Plus, you get to keep all the money you earn to spend as you wish. You don't have to pay for "services" you don't want. (Estimates are that everyone is about 8 times more wealthy than they would have been during the era of The State.) That means there is lots of money to spend on things such as museums.
Here in the free society, museums compete to attract the best scientists and researchers. This means those who pay well, or provide the best opportunity for the scientist's particular interests, can get the scientists they wish to have on staff. They also compete to attract paying visitors with engaging displays and innovative design.
Objects of interest are not likely to languish in the "collections" in the basement since someone spent money to obtain the item and they have a financial interest in getting it understood and described and displayed, rather than letting it gather dust and be forgotten. They also usually have an interest in getting the information right rather than in being sensational, since word gets out otherwise. But, that isn't always the case.
Some goofy museums exist, such as "Creationism museums", but few people take them seriously and see them more as a sideshow displaying freaks of sloppy thinking. Sure, they still make money but you don't support them unless you want to, just as you don't support a scholarly museum unless you want to.
It's similar to the way libraries operate here in Libertopia.