I'd think the menial job of being in charge of the bridge during a shift would be given to as many junior officers as possible, as part of their further training. It would only be on a rare occasion that the senior officers would bother to sit through these watches unless something interesting was going on, or unless they needed to get more "frequent flyer" points or whatnot.
So, Data would not mind doing the tedious task, as he could do a dozen other tasks simultaneously, including relaxing; Spock wouldn't be far behind in that respect. Worf might do it out of a sense of duty. Riker or Dax would do it if the helmsgirl was cute enough. Kim or Chekov would do it in hopes of finally getting a promotion. Sulu would do it on those weeks when professionalism was one of his hobbies. McCoy would do it only on Diane Duane's orders. And Scotty or LaForge would do it only if there was a chance to wreck the ship and thus get sent back to Engineering.