Each prompt edits the current state of the files. If you both prompt at the same time, the second edit can be based on files from before the first one finished, and it writes over it. It's not really made for simultaneous prompting.
What works for my team:
- One person prompts in Lovable at a time (yes, basically your WhatsApp rule, but make it a shared "I'm in" status).
- The other works locally on a branch via the GitHub sync, and merges through a pull request.
- Split by area: one owns dashboard, other owns settings.