What are the units in which a radio signal's strength is measured?
The E field has dimensions of volts per metre and the H field is amps per metre. For a far-field radio wave, the ratio of E to H is 377 (120\$\pi\$). Basically, if the E field received is X the H field is 377 times lower than X. 377 ohms is the impedance of free space and is measured in ohms because the "per metre" parts cancel out. The 377 ohms comes from this equation: -



The power is E x H so strictly speaking it's watts per square metre. The antenna has a certain "aperture size" measured in square metres so, a certain watts per sq metre hitting an antenna of a certain area liberates true watts into the terminating impedance. Imagine it like a sheet of paper being illuminated by a light bulb - the power received falls away with distance squared and this implies the individual E and H fields reduce linearly with distance.

You can use dBm or dBuV/m or dBI/m if you are talking about radio waves - whichever you use leads to the same conclusion and cross references the other units perfectly. However, you cannot sensibly state dBm without implying an antenna is present. You could state dBm per sq metre of course because that is the power illuminating a notional 1m surface at some distance from the transmit antenna.