The circuitry in a standard radio transceiver will probably draw around an amp while it is powered on - this includes the CPU, display and other such things that are always running. Handheld radios obviously try to keep this current as a minimum, because they run on batteries.
While receiving, there is also the power used by the receiving circuitry, and the audio amplifier, while there is audio being sent to the speakers. This can use an additional amp or so.
However, the real current draw is when transmitting, where most of the current is used by the PA (power amplifier) final stages. These usually operate in class A, AB or B, which are not very efficient - but offer good linearity, and thus low distortion. See the Wikipedia entry on Power Amplifier Classes. These amplifiers require large heatsinks to dissipate the 'waste' energy as heat.
Note that mobile and handheld radios that only transmit FM can use class C amplifiers, which are non-linear, since they always transmit at a constant power. These are much more efficient than class A, AB or B amplifiers, and mean that mobile or handheld FM-only radios use less power when transmitting, and require smaller heatsinks.