Messages in Activity Sharing send to wrong person (me)

With the release of watchOS 3, my wife and I set up Activity Sharing. I'm properly getting notifications from her Watch regarding her activity. However, when I try to reply or send her a message, I'm seeing two problems. First, the default replies on the Watch are usually (though not always) not the ones you see on this page http://www.apple.com/watchos/ , but are the regular default replies (Thanks, BRB, Hello!, etc.).

I might be able to deal with that issue, but the other issue is even worse. Instead of sending the replies to my wife, they are coming to other addresses that I have associated with my Apple ID. IOW, it's sending the messages back to me. Has anyone else seen this?

I just had my wife try to send a message to me. It worked, but it didn't give her the default Activity Sharing replies.

Also, I tried to send her a message via the Activity app *on my iPhone* and that was also going to send to me, not her.