In MN you have the following options
Purchase handgun or "scary" rifle from an FFL
#1 - show your valid MN CCW (which you had to take a class for, pay $100 for, wait up to 30 days for the initial check, and every year they re-check your eligibility) and get your gun immediately. While this is expensive, if you are denied the permit to carry, you can appeal and if your appeal wins the police have to foot your legal bill. You also get to skip the NICS check.
#2 - show your valid Permit to Purchase. Once a year you go through a background check and within 7 days the CLEO gives you a permit. There is no reason a CLEO couldn't give you one IMMEDIATELY but most guys take the full 7 days. By law they cannot have their own additional requirements, and must give you a yes or a letter stating refusal at the 7 day point BUT there is no teeth if they just don't do it. In theory you could take them to court and force them to, but that is far from cheap. Note you must still go through NICS check even after showing your permit to purchase.
#3 In absence of a permit, the FFL can follow the rule below AND can charge for it, but most of them just don't do it.
Purchase/Gift handgun or "scary" rifle from a private individual
#1 Bring to the police the names, gender height, phone number, and identifying info (driver's license number) of BOTH parties. Permission to allow police to dig into medical records. Signed and dated by BOTH parties. You must then wait 5 BUSINESS days for the CLEO to get word back that the recipient is barred and the delivery cannot happen. The law actually calls it a 7 day waiting period then goes about mentioning 5 BUSINESS days. If you don't hear anything, and you wait the 5 business days, you can complete the transfer.
P.S. CLEO can waive all or part of the waiting period if the recipient of the gun, or the family of the recipient is under some sort of imminent threat.
P.S.S Within 30 days of doing the above, you can apply for a permit to purchase and get it right away.
Now, in all actuality I think if you brought in this info and gave it to the police saying you were doing a private party transfer, they'd have no idea what was going on. I suspect that noone actually does the above. Of course by MN law all Glocks, well actually all polymer guns, are illegal becaue they are 'saturday night specials' because of their melting point.
#2 Show the non FFL your purchase permit and you don't need to do the transfer report
#3 show the non FFL your CCW and you don't need to do the transfer report