Analysing communication complexity of a protocol
(Revised based on poncho's comment.)

If p is of 120-bit length, then any field element should better be sent with 120 bits. If you want to send, say 1, with a 1-bit message, then this reveals the element's length and adds the computation complexity of constructing variable-length messages for different elements. Besides, only half of the field elements may have leading 0s, so sending messages with variable lengths does not save much communication bits.

As for the communication complexity, it depends on what message you are sending and may not be linear to the field size.