FYI for everyone having this issue. I am an Apple Watch nerd and have been testing every little thing with another equally as big Apple Watch nerd. I could get her sketches and was notified of them - taps and heartbeat too - but when I sent her something she never saw it or knew about it.
Just as mentioned in here, the recipient's iCloud email was not in the contact card. As soon as I added it and she added mine, it worked fine between us with notifications and all.
So it's definitely the iCloud email address that does it. You might use the phone number to initiate the contact and that's fine, but you need that iCloud address in the card to match because that's clearly what's being used for the digital touch and not the phone number. And that makes perfect sense since digital touch isn't a text message.
Here is EXACTLY what I did to fix the problem.
1. Go to the contact card of the person you want to exchange Digital Touch with on your iPhone.
2. Click edit and add the person's iCloud email address as a new email address.
3. Make sure you hit save.
4. Make sure the other person you are trading with also adds your iCloud address.
4. Reset/reboot your watch by holding in the crown and the button below until the apple logo appears. (not 100 percent sure this step is needed, but it definitely can't hurt)
5. Go to the Watch app on your phone and click on friends. Verify that the new iCloud email is showing on the card when you click the little letter i next to the name.
6. Try again.
Hope that works. Worked like a charm for me. Thanks to the kind folks in this thread who started the ball rolling. Hopefully writing it out in step-by-step form will make the process smoother.