Depends on the condition of the tyre, and the nature of the puncture.
If the tyre is in otherwise good nick with plenty of tread, and the puncture isn't near the sidewall, it can probably by easily fixed. As you probably know, the nail (for example) will be pulled out, then a rubber 'mushroom' plug is pulled firmly through the hole, and the excess on the outside nipped off. The air pressure, the friction of rubber to rubber as well as the centrifugal force of the spinning wheel will all work to keep the plug firmly in place. The tyre carcase is so thick and strong, and the hole so small, I don't see a problem.
The garage wouldn't attempt a fix unless it were safe.