I need to ask some questions before I can help you. Let's do this first. Opener #1 has 2 remotes that work fine. Opener #2 has no remotes. Is this true? How many buttons do the 2 remotes have? Do you want the remotes to open both doors? Look at the back of opener #2 just to the left of where all those white wires go and you should see a button. Is this true? Let me know the answers and I will try to help.

Don:
Yes, you are correct. Opener #1 has 2 remotes that work fine and Opener #2 has no remotes. Both remotes have 3 buttons, and yes, I would like to operate both doors with both remotes if possible. On the back of Opener #2, there is no button. There are the places where two white wires go (connected to Red and White) but where I would expect to see the button, there is only a circuit board. Any idea?

I am still a little confused. You say there are 2 terminals where 2 white wires go and you can see a circuit board. Could this be a small set of switches commonly called "dip switches.?

It has a series of about 15 tiny switches: 1 up, 2 down, 2 up, 4 down, that kind of thing. It's where the training button is on my other Craftsman GDO. Sorry, but I'm not sure what a "dip switch" is.

Aha, what you are describing are called "dip switches." This is an older opener and there is no way your remotes will work with it. I don't think the correct remotes are available any longer and I wouldn't recommend you purchasing them anyway due to the age of that opener.

OK, well at least I now know. Interesting. I guess we'll just have to break down and buy a new opener for this side eventually.

Thanks for your help!