Best Way to Integrate Subwoofers?

What’s the best way to integrate subwoofers with monitors, or speakers?

I’m not referring to placement, or room treatments. I’m referring to what’s the best way to integrate via a crossover, without a custom dedicated crossover.

Do you run two separate signals to the subs vs the speakers and supress the lower frequencies to the speakers, while supressing the upper frequencies to the subs? It would seem that this method might allow you to move the crossover to higher frequencies, relying more on the subs for the bass and lower mids - if that is beneficial. It would also seem that this method would permit you to taylor the slope of the crossover to minimize overlap of frequencies between the subs and the speakers, which might improve clarity - depending on the capability of the particular crossover used. And, I suppose it implies / requires a separate crossover to be used to run both signals through to route the mids and highs only to the speakers and the bass and lows only to the subs.

Or do you run the same signal to the subs and speakers and just emphasize the subs up to the bottom of the frequency range of the speakers, crossing over from the subs to the speakers at that point (crossover notch). It would seem that this method would require you to fix the crossover at the bottom of the frequency range of the speakers, wherever that may be, and would not require a separate crossover to be inserted into the signal path. And it would seem to imply that the slope of the crossover would be limited to the slope inherent in the subs and speakers, which would limit the flexibility of the crossover.

Or, does it make any difference?