My opinion: Your book needs to end with closure on the Trilogy Setup.
I think books follow the Act structure often used in film. Act I sets up the problem, introduces the characters, and ends with a clear statement of the Problem. Act IIa complicates the Problem, and ends with a turning point; Act IIb moves them toward resolution, and ends with the Key to resolution. Act III applies the Key, has the Final Conflict, and ends with the success or failure of the Heroes.
A Trilogy is the same, and the nesting of this principle: Each book has its own Act I, IIa, IIb, III. But the first book is an overarching Act I, the second book an overarching Act II, and the third book an overarching Act III.
The first book's problem is solved, but turns out to be the introduction of a larger problem.
The first book has to be satisfying, but the larger problem is what is going to sell the second book. It winds up with a satisfying conclusion to ITS 'smaller' problem but also delivers the turning point on the BIG problem. Wanting to see how that turns out is what sells the third book.
But each book, including the first, must have a satisfying resolution to its smaller piece of the puzzle, and yes, it is better if the time invested pays off in a BOOM.
Just like a single book, your characters for a trilogy should have an arc that spans the trilogy. Characters playing roles important to the outcome need to be introduced in Act I (the first book), or at least referenced.
Now if you have different characters in each book and there is no overarching story line or trilogy-wide character or relationship development, that is not really a trilogy (in my mind). The books are all just tied together by a setting; and it wouldn't make a difference if there were three or a dozen (like, we could write a dozen unrelated stories set in Middle Earth, it is a fun setting).