They don't necessarily have same premium, in most cases not.Think about call-put parity,C-P=S0-K*e^(-rt),the right hand of the equation is not necessarily zero.
When dividend is taken into consideration,by the same token,the answer is still no.the parity becomes C-P=S0-D-K*e^(-rt)

In general, their price should not be the same. But they can be equal in some special cases.

You can consider the problem mathematically according to put and call parity C-P=S-Kexp(-rT). You set C=P and get the stock price should be equal to S=Kexp(-rT).

Intuitive explanation can be a little complicated. Consider the near at the money option, call and put locally has the same insurance value around K. If r=0, at the money call and put should have the same price since they have the same time value and intrinsic value. If r>0, S should be a little lower than K so that to give put more insurance value to compensate the time value of money it lost due to the positive interest rate.

Consider the dividend, the situation is quite similar.

In short, the main point is that, their prices can be equal or not equal due to different factors. So there is not a general answer.
hope help~