The gardaí should only be called to deal with people refusing to wear face coverings "as a last resort", the justice minister has said.
Wearing face coverings in certain indoor settings became legally enforceable at midnight on Sunday which, officials say, should result in an increase in their use.
The Justice Minister has confirmed that gardaí will have ultimate responsibility for the regulation, which comes with a potential fine of €2,500 or up to six months' imprisonment.
Minister Helen McEntee, however, has said that she hopes that the gardaí will not have to use those powers and that the public will get behind the measure.
"The approach that we're taking would be similar to that of public transport," Ms McEntee said.
She said when people enter shops, there should be signage to encourage them to wear masks.
If people enter shops without a masks or remain in a premises without one, she said she would expect some engagement between the customer and the retailer.
"As a last resort, that An Garda Síochána could potentially be called," Ms McEntee said.
"We have, throughout this entire pandemic, tried to engage with people and ask them to comply with the regulations that have been set out. I think, in the vast majority of cases, people have done that and if we look at transport and the introduction of face coverings, there has been a huge level of compliance. It's only in a small number of cases that the gardaí have needed to step in.
"So we hope that this will be the case. And I think, already, you can see and I have seen the huge levels of increased compliance of people wearing face coverings. It's not 100% and we need to get to that point. But obviously we want to do so with the support of the general population and that's the intention."
Under the regulations, which will be in place until at least October 5, a face covering is defined as "a covering of any type which when worn by a person covers the person’s nose and mouth" and must be worn in almost all indoor retail settings, save for the post office, credit unions or banks, eating and drinking at a restaurant or café or undergoing medical or dental services.
Special dispensation is given for what the statutory instrument calls "reasonable excuses". These include people who are medically unable to wear masks without severe distress, those who would have difficulty communicating or those who are providing emergency assistance. Shopworkers can also ask people to remove their masks temporarily in order to verify their age.
Gardaí are also exempt from wearing the coverings.
Retail staff must also wear a face-covering unless there is a partition or they can stay two metres from others at all times.
Retailers say that the rate of wearing face coverings has gone up in recent weeks.
Duncan Graham, managing director of Retail Excellence, said, "Since the Taoiseach announced, over two weeks ago, about the compulsory wearing of face coverings in retail, compliance has steadily improved. We take our responsibilities seriously and will be looking for the support of the public to help make shopping safe for all."