My recommendation (if they were my thermometers) would be to somehow get documentation as to what their stated accuracy is. Depending on the particulars, liquid-in-glass thermometers are marketed under many different brand names. First would be to get a copy of the original manufacturers documentation. If you can't do that, you might get an Omega Engineering Temperature Handbook, Fisher Scientific (those are the two that most immediately come to mind; no intention to try to sell their products). Find a stated accuracy for your thermometers, make a copy of the page and retain that documentation. I think it important to have a document on file as to what their stated accuracy is (I will respond to your comment about creating your own specs later in my reply).
As for the certificate from the contractor.Thermometer Manufacturer I have been a lab supervisor as a contractor doing calibrations. As far as I am concerned, they MUST state in their documentation what the tolerance limits that they calibrated your thermometers to. It may be that you have to pay extra to receive that documentation. But it is a must. My preference (and implicit in some standard thermowell shox r4 requirements) is to have any or all cals done with BEFORE/AFTER data (in this case, only AFTER, as you cannot adjust liquid-in-glass thermometers). On the data sheet must be stated what measurement points are tested by the contractor, and must list the upper and lower tolerance limits. In order for a calibration (in this case, only a verification) to be valid, you must be checking to see if your thermometers are measuring within a specified set of tolerance limits. The contractor then must use standards that meet acceptable limits of uncertainty (for example a drywell(dryblock) calibrator, or other).

You mentioned that the contractor just recorded readings at different temperatures. Did the contractor list tolerance limits? And did the contractor Pressure Gauge Manufacturer state what measurement standard was used? Both of those are a must. If the contractor does not state tolerance limits for your thermometers, he/she cannot fairly state that they passed calibration (verification).

As for your question about establishing your own accuracy, using one as a standard..... If you don't have an established calibration program, and don't have trained, skilled people to administer it, without getting into a lot of detail, my gut feeling is that you could get yourself into trouble. You would have to basically document a whole program to be able to legitimately do that. Additionally, you would have to assure that the thermometer  cheap puma shoes you use as a standard is well documented, nike air max characterized, capable of the desired accuracy, and you would need to purchase some standards (if you don't already have them). You would also minimally need to be able to place the standard and the thermometer being tested in an environment uniform and stable enough to compare them. An Ice Bath can be wholesale ugg pretty economically produced. But you will also need to be able to create a uniform elevated temperature (preferrably at least two temperatures across the range of the thermometer in addition to the ice bath). There are a lot of variables that would need to be considered.