In many states, a contractor must have the appropriate license to pull a permit but there are exceptions that allow owners to pull their own permits. If a homeowner pulls their own permit, if the inspection reveals something does not meet code, the homeowner bears the responsibility for the changes to meet code. If the contractor pulls the permit, the contractor is responsible. Good contractors know the code and can work with inspectors. Unless you know what you are doing, the contractor should pull the permit. If the contractor asks the homeowner to pull the permit that may be a red flag that the contractor does not have the required license to do the work.
While you may "save" a few dollars by not pulling a permit, you risk finding out that the work does not meet code, you will be asked at resale if there was unpermitted work on the home, and you may bring down the value of your home. Your "savings" may be very costly in the long run.
I'd love to hear from others on their response to consumers who want to avoid permits or who indicate that they have other bids that dont include pulling permits.