Criminal record check is completed by both countries. They scan your passport and anything undesirable shows up on the screen. I know as I used to do to work as a customs officer.
Not every criminal offence will automatically keep you from crossing the border. For instance having a minor offence like disturbing the peace, that happened 40 years ago shouldn't be an issue, but it might be and finding out while waiting in line for clearance is not what you want to happen.
Anyone that has an offence should contact the embassy of the country they wish to visit and see if they are eligible for a pre-clearance.
It really depends on how serious the offence was, how long ago it happened and if you have been a clean living citizen since.
It's all about the risk that you pose
