The function hasn't been removed, it's been moved.
Go to the profile page of a shared individual (same person in both private trees)in your tree. Hit the Member Connect bar, the data of the individual in the other private tree should appear, whether or not you've previously connected the profiles. You can copy any data, including images, from the profile in a private tree you're invited to, to yours.
>>As an aside, Member Connect is not available for living people.<<
There may be limitations depending on what status your tree invitation is, ie, whether you've been allowed to view living people.
>>I can click on the "add this person to my tree" and a list of my trees comes up, but when I try to select the tree I want, it will not let me. <<
This sounds like a browser issue, try another browser and see if that doesn't clear up that issue.