It sounds like you have a second bilge pump to drain a compartment that does not have limber holes and, hence, does not drain to the bilge on it's own. Discharging that compartment to the bilge, to be picked up by the main pump, is perfectly acceptable. It's more a housekeeping thing than anything else, if I'm reading you right.
The cockpit pump, I would imagine, does not dump the water into the bilge but into the cockpit where the cockpit drains would discharge it overboard. Why anything pumped or drained from the cockpit, or cockpit locker, should go to the bilge is a mystery to me, and probably a sinking in waiting. What does the cockpit hand pump suck water from? What type of cockpit drains do you have? And where do they drain to?
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