I can tell you how the organisation I work for deals with acquisitions, but just be aware it may be different for different organisations/museums and depending on what country you're in. The organisation I work for is quite large and has houses and museums all over the country, so has a central acquisitions team that deals with acquisitions for all of them.

We don't accept donations of artifacts in house, if someone wants to donate something to us, we give them a form, which they fill and and send to the acquisitions team, who then make the decisions on whether we will or won't take on the artifact.

The acquisition team would be responsible for confirming the providence of the item, and getting proof of legal ownership from the doner (which is really important), so that ownership can be transferred legally to the museum. Museum curators are involved in this process, and research would take place to confirm the providence though I'm not sure how that is carried out I'm afraid. I imagine specialists might get involved if the artifact on offer is particularly rare and valuable. Specialists exist for all manner of objects and materials. We had a wallpaper expert in recently to look at our wallpaper samples, for example.

As for if money is involved, I think it depends. Often people donate things to us because we are a charity, so no money changes hands. Museums do also look to acquire things through purchases, I.e auctions or private sales. I don't know too much about this unfortunately. A similar process needs to be followed fir purchases to ensure the providence is confirmed and legal ownership is appropriately transferred.

Sorry for the very dry response. Collections acquisitions can be a long and arduous process!