Things have probably changed since my time, but (and I will qualify this with it was a while ago), DWO's are responsible for the safe navigation and operation of the vessel when underway. They are the CO's representative on the bridge when the CO is not up there, and can have very varied responsibilities granted by the CO as to latitude in changes in vessel speed, communications, course changes, prosecuting targets, managing other personnel on watch, responding to casualties (engineering and personnel), and just about anything else related to the operation and safety of the cutter (including Un-Rep and Flight Ops). In port the junior DWO's will likely also stand in-port OOD meaning they are acting on behalf of the CO when in port, and usually have control over all other watch standers in port and are responsible for every aspect of the ship while on duty, and are the leader of the "Duty Section", which depending on the size of the cutter can be a lot of different people performing a lot of different duties all over the ship. Both DWO and OOD are Highly Responsible Positions, and you need to know your people, regulations, and Captains Orders or you can get in trouble rather quickly.
Junior Officers also will also be assigned "Collateral Duties" which can anything from Public Affairs Officer, to Laundry Officer, to Morale Officer and so forth. As Junior Officers they will likely (if on a Law Enforcement Duty) be assigned as Boarding Officer or part of the Boarding Team, boarding vessels in different weather conditions, day and night, and in high stress situations.
Usually afloat tours are for two years (at least they were). Depending on the type of cutter you could be underway for less than a week at a time or three months at a time and anything in between.