Port police said 48 survivors managed to reach the shore of Farmakonisi after the first boat hit rocks around 0030 GMT as it was making the perilous crossing from Turkey.
The coastguard recovered the bodies of six children and a woman.
Several hours later, a second boat capsized off the small Aegean island of Kalolimnos. Port police recovered eight bodies, with the 26 survivors reporting that dozens of people had been on the boat.
A search was under way for the others, the port police said, AFP reported.
People fleeing war and misery in the Middle East and elsewhere -- many of them Syrian refugees -- are still arriving from Turkey in flimsy boats in their thousands every day, despite the dangers and the harsh winter weather.
The International Organization for Migration estimates that some 31,000 migrants have reached Greece by sea so far this year, hoping to start new lives in Germany, Sweden and elsewhere in the European Union.
The IOM reported 77 people killed trying to cross the Mediterranean between January 1 and January 17.
On Thursday at least 12 migrants, including children, drowned off the Turkish coast as their boat tried to reach Greece. The Turkish coastguard rescued 28 people.