In regards to temperature rise, it largely comes down to two factors.
1) How much heat is generated by current flowing in the wire.
2) How well the heat can be carried away.
Assuming that you're frequency is comfortably below the point at which the skin effect comes into play, the heat generated is mostly I^2*R.
Getting rid of that heat is a more complex activity. A piece of magnet wire in free air can carry more current than an equivalent thermoplastic insulated wire.
1) The insulation is thinner, so it provides less thermal insulation.
2) The insulation can operate at a higher temperature.
But that is not the only thing to consider. Magnet wire is seldom used in free air. Instead you wind a bunch of it in inductors and transformers. Now you have changed the thermal situation significantly. Each turn is not in free air, but surrounded by other wires that are also pushing heat out. Figuring the current capacity for the winding of a transformer, is basically looking at how much heat x inches of wire will dissipate, then factoring that into the volume of the finished coil.
But remember:
1) The inner layer will be hotter than the outer layer.
2) The resistance of copper increases about .004% per degree C of temperature rise.
3) In a transformer with multiple windings, the heat will be the sum of the heat in ALL windings.