The ones I'm familiar with flow water whenever the supply pipe has a pressure drop of 5 to 10 psi. That is, if the trap primer is fed from a pipe leading to the washing machine, it will open every time the washing machine fill valve opens. This means a small squirt of water happens a few times a week in this application.
I don't like trap primers because they fail after a few years, they're not something that gets inspected so when they fail, it's as if they were never installed.
So it probably does drip more than it is needed but it does not run continuously.
Other tricks to solve problems with dry traps: a few drops of baby oil in the trap will make a film over the water in the trap and significantly slow evaporation.
Or install a "barrier type" drain trap into the drain. These are also called "duck bill" valves and use a flat rubber tube that opens whenever there's water running into the drain and closes against sewer gasses when there is not. Not as good as water in the trap or for use as a substitute to a trap but they've solved a lot of sewer gas smell complaints for me, especially in hard to access air conditioner condensate drains that dry out in the winter. These barrier traps just get pushed into the drain opening with some caulk to hold it in place and seal it.