Basically, yeah, the private document does work, but there's an important nuance that several people are overlooking. The paper you two drew up is legally valid, but that doesn't mean it's enough for everything. When you go to the traffic registry to change the vehicle's paperwork, they're going to check that it's properly signed, that it has the correct information from both of you, the license plate, the chassis number... all that. If you did it carefully and everything's clear, they'll probably accept it without any issues. Now, the thing is that if something goes wrong later (a complaint, a financing problem, whatever), a private document between two mates is way weaker than one notarized. I'm not saying anything dodgy happened, but legally speaking, without a notary there's less protection for either of you.
My realistic advice is that you check directly with the General Traffic Directorate in your province or your local vehicle registry, because even though most places accept the private document, some ask for more specific stuff. Some registries tell you to just bring the paper as is and you're done, others ask you to have it certified, others ask for a specific contract template. It's half an hour on the phone and you'll be sorted. If for whatever reason the registry says it's not good enough for them, then you'll have to go to a notary, but I'm pretty sure they'll accept it. The important thing is that the document is done properly: full names, ID numbers, correct license plate and chassis number, the amount of money, date, and both signatures clear.