Just over 4,000 calls were made by the Border Management Unit (BMU) to check up on almost 62,000 passengers arriving at Dublin Airport during the first two weeks of July.
It follows a parliamentary question from Social Democrats co-leader Róisín Shortall, which shows just 52% were answered.
A sample of around 6,100 passengers was taken, with around 2,000 exempt from follow-up as they were transiting to Northern Ireland, or in the State for less than two days.
At Cork airport, 1,359 calls were made and 720 were answered meaning 53% of follow up calls were answered.
Sinn Féin leader Mary Lou McDonald says further quarantine measures are required and that this needs to be monitored and effective.
Ms McDonald said Ireland needs "some form of effective monitoring, effective checks and balances, and effective quarantine," as the measures in place are not enough.
"Currently, people land, they fill in a form, they may or may not get a telephone call, lots of people are now reporting that they get no contact whatsoever," she said.
The latest travel quarantine figures follow the government's announcement of a green list for arrivals in Ireland.
15 countries were published which the government considers safe enough for travel so as people returning or arriving from these countries will not need not to restrict their movements.
Travel insurers say they will provide cover to anyone visiting the 15 countries.