Depends on the size and type of vessel. On large, deep sea vessels (and perhaps most vessels of any kind) while the captain has ultimate authority over the actions of the chief engineer, in a strictly legal sense, the chief engineer actually operates autonomously, only answering to the captain for the broadest of orders. For example, the captain tells the chief when he wants the main engines started and stopped. But the captain wouldn’t tell the chief engineer how to do start or stop them, or who to use to start them, or how to perform maintenance in between. On the largest of vessels the chief engineer is nearly co-equal to the captain in authority. The Chief runs the engine room and engineering plant without orders or counsel from the captain, except to start and stop a certain piece of equipment. On the largest of vessels the chief engineer manages a staff of people very nearly equal in size to that of the deck department, without interference from the captain.
But as the size of vessel decreases this becomes less and less the case. At the opposite end of the spectrum, on smaller tug boats for example, there may very well be no “Chief”. No licensed engineer at all. A crew member may be designated as “engineer”, but the captain may make all the decisions about operating the engine room, when maintenance occurs, etc.
In the U.S., the status of the chief engineer on most vessels falls in the middle of these two extremes. Certainly, if there is a licensed chief engineer aboard, most captains do not interfere in the running of the engine room. To most captains the engine room is an enigma over which they have little or no expertise. They are only to happy to let the
"chief" alone.