Great question. Yes, they were allowed to fish, albeit on a limited basis.
Last year, I read a new book from Naval Institute Press, called The French Navy in World War II. There's a brief mention on page 304 of what you're interested in, so I'll quote directly from that source:
"Some thirty trawlers and sailing craft, caught by the armistice off the fishing banks of Newfoundland, sailed to Africa to continue their duty of feeding the population of France. The fishermen of the Occupied Zone were permitted by the Germans to continue their fishing, but only in small vessels with small crews, and restricted to areas just a few miles off the coast. A German patrol boat generally kept guard over the fishermen, though sometimes a squad of German soldiers would take passage on the fastest boat in the fleet to maintain its surveillance from there. Nor was the Kriegsmarine wrong in distrusting the fishermen of France. Often a fishing boat, slipping past the German patrols at tremendous risk, would carry to England a load of young Frenchmen anxious to reenter the war, or Allied aviators shot down over France, or ─ even more dangerous cargo ─ secret intelligence agents!"
This is, of course, a rosy view of the situation. French fishing boats were the targets of Allied air raids, submarines and were the innocent victims of mines laid by both sides. Mines in particular were a serious threat; they could break loose from their moorings and be struck by fishing boats. They could be pulled in the nets of fishermen and set off that way. "By November of 1942 the number of fishermen killed or missing amounted to 160, and grew to several times that before the end of the war," The French Navy declares.
The principal instrument of German control with regard to the French fishing industry was the Marineküstenpolizei, contrasted with the Wasserschutzpolizei, which principally operated within Germany itself. The Marineküstenpolizei were established with an order dated Sept. 9, 1941, but I've also seen a published reference to a unit deployed in Denmark in 1940, which I can't immediately reconcile. In any event, the WSP and MKP seem to have been structured very similarly, frequently trading staff.
To keep the fishing vessels under tighter control, fuel restrictions (both deliberate and through wartime shortages) left them with limited range. This made long cruises more difficult. Furthermore, fishing vessels typically were operated by men with family ties on shore.
British trawlers continued to operate in the North Sea and the Atlantic. They were vital to Britain's food supply. The Royal Navy also requisitioned trawlers as mine sweepers and armed boarding vessels. Probably about half of all trawlers were lost during the war. Many trawler crewmen joined the navy. I cannot think of any books abut almost every fishing port in Britain will have a website about the trawler fleet. Look for ports such as Harwich, Doverport, Grimsby and Scarborough.