Modem power usage versus signal strength
Indeed, GSM devices in general change their transmission power when receiving a weak signal. The technique is called transmission power control, or just power control.

I'm also 99% sure that they change the amplification of the receiver.

Note that when receiving or making a call, the transmitter will start at higher power, to lower it afterwards when the connession is established. That's why some people advocate waiting a second before taking it to the ear.