How does one sign a check over to another person (in the United States)? 

I believe the person who is named on the "Pay to the order of" line
must endorse the check on the back, as usual. But, I'm not sure of the
rest of the procedure. Does the endorser have to write "Pay to the
order of..." below his or her signature, followed by the name of the
next recipient? I have heard that this is not needed, and that the
next recipient merely needs to sign his or her name under the first
endorsement.

That sounds like a bad way for banks to do business. Shouldn't there
be some written proof that the first endorser meant for the check to
be given to another party?