I shall answer your questions.
1. Is a menu supposed to appear when Share is pressed? If not, what does this button do?///
The 'share' button only works if you have saved your mindmap in 'Mindmap Public Server'. Otherwise it has no meaning. If you have saved your mindmap in 'Mindmap Public Server', then a window will appear to share your mindmap.
From this 'share' menu, you can share your mindmap to other people via 'Facebook' or 'Twitter' or 'Google Plus'.
You will also get 'Permanent Url' and 'Short Url' to your mindmap. You can use these urls to share your mindmap to other people publically.
2. By pressing this button, does this mean that my mindmap is now publicly accessible/searchable? ////
No. Your mindmap cannot be publically accessible/searchable. This button is used to share your mindmap to other people via Facebook, Twitter, Google Plus or via direct url.
2(a). If so, is there a way to make it private again? (my apologies for pressing a button I did not understand!)
Your map is private if you have saved your map in Google Drive. If it is saved in Mindmap Server, then it is public. However you need the URL to open the mindmap. Without the URL, none can access it even if you have saved the map in Mindmap Server.
Hope your query has been answered.
Thanks