There will be times when you may be on a call and another call comes in. It is ok to explain to the client and have them hold while you pick up the other line. When you pick up the other line, politely ask them to hold and go back to your original call. Finish up the call, open the ticket as needed and switch back to the call you had on hold.
Both dispatchers need to work as a team. If only one dispatcher is taking on the main load of calls, that is unfair to the other person. If both dispatchers are helping each other, there should be very little times when you have to put clients on hold.