Some 50,000 migrants are now in Greece and more arrivals have been reported.
The agreement between the EU and Turkey is aimed at halting the influx of more than a million "irregular migrants" through Greece since January last year.
Prime Minister Alexis Tsipras has said Greece faces an "uphill effort" in implementing the deal.
Boats are continuing to arrive from Turkey. More than 1,600 new migrants have reached the islands in the 36 hours since the measures came into effect on Sunday, officials say.
"If there is no reduction in the flow [of refugees], we will not be able to evacuate the islands successfully so the deal can start to be fully implemented,`` Mr Tsipras warned
The arrival of the 10 Turkish monitors has aroused some political opposition in Greece. They will stay on the islands for a week.
Meanwhile, 2,300 EU experts, including security and migration officials and translators, have not yet arrived to help enforce the plan. Officials have said the returns to Turkey are unlikely to start before 4
April.
Under the deal, no new migrant arrivals will be allowed to travel to the Greek mainland. Anyone who does not apply for asylum will be sent back, as will anyone whose claim is rejected. For every Syrian migrant sent back to Turkey, one Syrian already in Turkey will be resettled in the EU.
There are also questions about what will happen to the tens of thousands of migrants still stuck in Greece and on its closed border with Macedonia, as their route north has been blocked.
Boats carrying some 2,000 irregular migrants from the islands docked near Athens on Monday.
Migrant crisis: Turkish monitors on Greek islands for EU deal
22 March 2016
Read: 870
Turkish observers have arrived on the Greek islands of Lesbos and Chios to oversee an EU deal on sending migrants back to Turkey.
-1780923278.jpg&h=190&w=280&zc=1&q=100)
