It’s published by the KM Group (KM standing for Kent Messenger; the group rebranded itself in 2008). The editorial offices are in Strood in Rochester.
Although a Medway Towns edition of the Kent Messenger was launched as early as 1968, the Medway Messenger itself didn’t start publication until 2003.
It comes out on Mondays and Fridays and is thus the KM Group newspaper with the highest frequency (all the rest are weeklies). It’s also available by (paid) e-edition online.
There’s an associated freesheet, the Medway Extra.