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Note that your trade power will not be combined but rather the privateers tp will go to privateers while any other will go to collecting.
In some specific cases it is possible to make more money by privateering your own home-node, since you auto-collect there, but that's generally only worthwhile if your share already sucks, since then you'll squeeze out a little bit of extra share due to the bonus tradepower of privateers, and aren't trying up a merchant to do it with. But that typically becomes counter-productive outside of the very beginning of the game as an OPM, since once you start expanding, you'll typically cross the threshold point where protecting becomes more valuable.
Generally though, using privateers is simply an alternative to collecting with a merchant, except you trade potential diplomatic consequences and ship upkeep instead of losing your transfer bonus and tying up a merchant. The pirate payout vs loss of 50% trade-power is a bit harder to quantify.
Pirates pay 40% of what they collect. Non-primary merchants lose 50% trade power if they collect (plus the loss of transfer bonus to your capital as well as oppertunity losses for having the merchant transfer trade from somewhere else to your capital nodes).
ex. Ottomans in a massive war, then privateer all their nodes. The AI will more than likely debt spiral.