Faucet handles fell off!

I just purchased a new home and found that both my hot and cold handles to my bathtub have no screws to set them onto the valves. As a result they just lift right off. I can pop the little cap off of each of the handles and I see that it has a threaded interior shaft. I can see that the valve stems are in fine condition, but the hole in the middle is not threaded and has a much larger diameter hole than the threaded one in the handle. I thought that it would be as easy as finding a new screw to lock these handles back down on the valves. Is the thread way down inside the valve stem?
Here is a similar faucet to help understand what I am describing.

Also, I see that the sink has the same handles so I popped the little caps off of each of these handles to see what the screw looks like. To my surprise the screw doesn't even have a head. It looks to be nearly flat...not allen or hex. So it doesn't even look as if it can be turned by a tool. It turns out that the sink coldwater handle is a bit wiggly too.
I am at a total loss here... Thanks.

OK, I have been doing some online searching and I think that I am getting closer to the answer. I think that the handle and the escutcheon are supposed to be affixed together (this doesn't explain why my handle fell off) but what is missing, and now makes more sense is that on my sink faucet, I'm not looking at a screw with no head...I am looking at the BOTTOM of a screw! I am going to attempt to unscrew the entire escutcheon and handld from the sink tomorrow. As for the bathtub faucet, if I am correct, then the upsidedown screw is just plain missing.
So now I have 2 questions,
1) how does the handle affix to the escutcheon
2) and maybe related, where does that upsidedown screw mount in order to screw into the handle from below?

This is an example of one model ( Moen Monticello) that works as has been described: http://www.moen.com/shared/pdf/T4570pt.pdf

My faucet looks more like this,
http://www.moen.com/shared/pdf/T6125pt.pdf

I think that I have come to the conclusion that the screw has backed itself out and, in the case of the bathtub faucet, fallen down into the valve stem.

Am I correct in thinking that I have to unscrew the bell shaped base (escutcheon), pull out the adapter piece that slips onto the valve stem, and find the lost screw down there. Upon which I can then rescrew back together the bell housing (escutcheon), adapter, and handle? If so, I am definitely using locktite on this screw so this doesn't happen again.