Can mac receive messages when iCloud keeps asking for password? One day my older Macbook Pro (which is now not in my hands) started asking for iCloud password with text "This Mac can't connect to iCloud because of a problem with mail@mail.com." but I kept postponing it. The device is running El Capitan.

The deal is I don't want the person who was the mac in their hands now (I'm not sure I had a log out after 5 minutes set to on) to read my messages and I was hoping that this would mean they simple would not receive them on the mac - because it kept asking for my iCloud password.

And a tiny subquestion #2 - can I check to see when was the device last logged in?
If there was a problem with your iCloud account, and it was requesting your password, no, they shouldn't be able to read your messages.

However, to be sure, I would check with the current owner.

Always follow this guide before giving away your Mac.