I think they shouldn't since they are relatively at rest to each other. The actual answer to this question (it was in an exam I took) is that there will actually be a magnetic force acting on both of them.
Here's my reasoning: Since both the charges are unaccelerated, I can view them from an inertial frame which is at rest relatively to both the charges. Now, these two charges would appear to be at rest and as such they should only affect each other by electrostatic force. What is wrong in my reasoning?