Depending on which hospital you go to, you will get different answers. My dd1 swallowed a coin the size of a 20c piece at age 4. For her X-ray through gp and it was in bowel. A week later we had to go to ed and were told to go away. Dd developed signs of a bowel obstruction a few days later so went to another major hosp who told us the coin had refluxed. A month later, after lots of symptoms, a coworker pulled strings and got us an appt with a paed gastroenterologist who removed the coin two days later.

So yes, always get them checked but don't be surprised if you get turned away. Especially for small coins.
Usually anything the size of a 10c piece or smaller will pass easily.
If they are coughing and spluttering always assume a blocked airway and tell emerg that. An X-ray is a quick way to determine that.

Hopefully your little one passes the coin soon! We nicknamed dd1 the human vending machine after her incident.