I haven't used it but MSP has had more than one testing site, with different conditions, at MSP. This is all T1; I don't know anything about T2.

At times, there's been testing in the parking ramp, but this might be just for airline and airport employees.

I've seen signs to something on the level below arrivals, where the tram runs to the light rail station and the back of the parking ramp.

I've also seen signs and an information page (hard copy at the airport information stands staffed by volunteers) for an airside testing station at the beginning of the F concourse, toward the left and before the first gates as you enter F from the TSA checkpoint or from the main mall with many merchants. These tests are not free, although as expected antigen is cheaper than PCR. They claim to be fast but they're located after the check in counters, so having documents to pass TSA airport security could be a problem..

BTW, the saliva tests done in MN seem to be from Vault. They mail out a kit for a saliva test that you can do at home with supervision over zoom and then drop off at a UPS store for shipment to their laboratory. They're not instant tests, nor are they free normally AFAIK.

There's usually testing done at MOA or in one of its outdoor parking lots. MOA is several stops by light rail train from the airport, with tickets roughly in the $1-3 range (by from a machine before boarding), with service about every fifteen minutes at normal times.

AFAIK, by policy MN doesn't insist on ID or state residence for either testing or vaccination for COVID-19, but of course, they'll take your information and bill your insurance if possible.

ADDED: A very expensive alternative would be to use the travel medicine outfit at HCMC. They claim to verify testing (and vaccination, etc.) requirements for travel and then give results in time for travel. This is likely to cost about $500 and not be covered by insurance.

Have you tried calling the airport to ask? When I once tried to check, I couldn't find anything on the airport's website, but the testing locations seem to constantly change.

I don't know if it's still available, but Minneapolis Convention Center (downtown) has been doing the Vault saliva tests in person (just as Vault served the community vaccination site there when it was operating last winter).