Is drilling through the side of the tooth by mistake a common problem with root canals?

No, it isn't a common problem. Endodontist, or whoever, should know his business better then that. That can cause mega problems.

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What kind of problems?

Have you had a root canal, and have a specific problem. Please give more information.

If the wall of the tooth is breached from the inside, then infection can proceed into the bone around the tooth.
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I had a root canal started some months ago, then the dentist sent me to the specialist I don't know why, and they quoted me a price I could not afford, so I found another dentist who could pull it. That dentist then offered to try to do the root canal for the same price the first dentist quoted. So yesterday he opened up my tooth and told me the first dentist I had gone to had veered out the side of the tooth by accident. And that the tooth would have to be pulled.

It could be have been a potential problem then. You could have had a really sore mouth, with infection going into the bone. It is a good thing the tooth is going to be pulled, if that is what you want to do, however, it can be fixed, there is a compound they use to repair the breach in the tooth but costly.

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This kind of thing can happen but doesn't very often. The tooth WILL have to be removed eventually as the perforated root can get infection around it. It may have to be removed even if the dentist did repair it. I would recommend that you see an endodontist for something like that as they have more experience with it. The specialist (endodontist) would also be able to tell you what kind of prognosis the tooth has. You should be able to schedule a consult only appointment with the specialist and take whatever x-rays your current dentist has of the tooth with you to save you the cost of having new ones taken. (Hopefully they are good quality xrays)