I agree with all the other answers: you will never get compliance with your current system. There's zero incentive for people to participate in reporting their peers, and being reported isn't the best motivation to keep your screen locked anyway. You need to ditch this plan.
Some comments suggested you start sending out emails from unlocked computers offering to buy the whole office breakfast. Some others said this is a bad idea, because it will make people really upset. I agree that it's likely to backfire if you do it to an unsuspecting victim, but I think there's a way you can make it work.
Get a big wig (like your CEO or someone like that) to play along with pretending they left their computer unlocked, and somebody else sent out a "I'll buy everyone breakfast on Friday," email. Have them send it out themselves, then follow it up a bit later with something like, "Whoops! I forgot to lock my computer when I got up, and [chief of security or whoever] sent out that email to teach me a lesson. I guess I will have to bring donuts on Friday for everyone. It's lucky that's the worst that happened, when [some other real and serious threat, like "hackers would love to steaI our credit card database"]. I know I've learned my lesson, and I hope you all have, too."
This serves multiple purposes:
It shows executives are being held accountable, and that they're serious about the policy
It demonstrates that there are real risks associated with leaving your computer unlocked, instead of some vague boogeyman
It gives people a reason to talk about the policy in a positive manner (they will probably think it's hilarious)
As a bonus, now you can periodically walk around the office and look for unlocked and unattended computers, and slap a sticky note on them that says "You must really want to buy the whole office donuts! =) " It's a consequence that's not too embarrassing, but it reminds people of the dangers. For repeated offenders, you can escalate by changing their wallpaper to be a box of donuts and things like that. Hopefully people will band together and lock computers they see unattended and/or give gentle reminders to each other.
Why do I think this will work? Shortly after I started my job, I was told a story of a co-worker who liked to send emails from unlocked computers offering to buy everyone lunch. I'm not convinced it's even true, but it got me into the habit of locking my laptop. I can also repeat the anecdote when I see other new hires being sloppy about locking their computers, and it doesn't come off as a lecture. I would never report my co-workers, but I have zero qualms about "protecting" them from a "threat" we both face.
In addition to that, I recommend you also configure all new computers to automatically lock after being idle for five minutes or so. Maybe also go around after the "free donuts" email and offer to set it up for people, so they "don't fall victim to any prankster-prone co-workers". It sounds like your employees are a bit protective of their computers, so make it voluntary, and let them know they can increase the time before it locks if they find it annoying; the configuration change is purely for their convenience.
You will never get 100% compliance, but these steps should set you on the path for improving your culture and making it more likely people will care about security. That's really all you can do without upsetting everyone.