No, it does not transmit continuously. In fact, it may not even necessarily receive continuously. One major point for optimization in phones is power efficiency. The phone may have to transmit with quite a bit of power (on the order watts, unlike wi-fi which is usually less than 0.1 watt) so running that transmitter continuously is out of the question. Even when you are browsing the web or performing some other activity that uses data, the phone will turn the transmitter on only when it needs to send data to conserve battery life. The low noise amplifiers in the receiver also consume significant battery power, so turning those off when not required also saves power. Obviously, there will need to be occasional communication with the cell tower to manage hand-offs, calls, alerts, etc. but this can be done 'relatively' rarely - say, a few ms every 10 seconds to minutes.
When transmitting, the cell phone tower actually tells the phone how much power to transmit with. This serves two purposes - since the phone uses exactly as much transmit power required to reach the antenna, it will save battery power. Additionally, and more importantly, the signal from the phone at the tower's receive antenna will not drown out the signals from other phones that could be much farther away from the cell tower.