Tower Controller reported a potential ground conflict when coordination with a adjacent controller was flawed immediately after the local control positions were de-combined.

Narrative

I had just taken over the position and the relief briefing was given to De-combine the position's. I took over one Local Control position. When the hand off of the positions was made I noticed the previous controller did not switch the aircraft to the appropriate frequency after giving them hold short instructions. I then told the controller next to me; now that the position was officially mine; that Aircraft X needed to be switched. I did not hear an acknowledgment or any other comment. There was Aircraft Y rolling down the runway at that time and as Aircraft Y became airborne I started to cross Aircraft X when I noticed he was already moving across my runway. I asked the adjacent Local Controller if they were crossing them and they said yes and thought I said they could. I said no I asked you to switch their frequencies. The other Local Controller had not done any coordination on the land line. I had no other aircraft in on my runway or lined up for my runway. There was not a loss of separation; it was just not properly coordinated. Land line is always safest. I know now I won't do that again. I'll keep it to myself and tell my fellow controller instead. Recommendation; I recommend everyone use landlines for coordination to cover their own butts; or to make sure they get a true acknowledgment from the other they are coordinating with. Not just a shake of the head or a 'hmmm' out of their mouth. An actual read back of 'yes you can cross Aircraft X after my rolling CRJ'.