My P also lets staff choose where their kids go. It's interesting to me, because she does NOT take requests otherwise, and in fact is adamantly against it. She also doesn't allow previous teachers to specifically place students with next year's teacher (they make the class lists, evenly distributed, and then P assigns a teacher to each list) because "we should be confident that every teacher in the building is a good teacher."
Most teacher parents who bring their own children to our school don't live in district, and I think P considers it a huge compliment that they bring the kids to our school, rather than attending the higher SES schools in their own neighborhoods. So I wonder if it's sort of a ploy to keep those kids around.
A previous P did take requests (from anyone) because she reasoned that parents who made requests were often high maintenance (I'm in a low SES school, so this wasn't the norm/it wasn't like there were 50 such parents). When they complained later about the teacher they requested, she could say, "Remember, you requested Ms. X" as opposed to the parent railing all year that xyz issues could have been avoided if their child had been placed with the teacher they requested.
If our current policy has caused any issues, I'm not aware of them. I personally would be nervous about having another teacher's child/wouldn't prefer that extra stress. So it's sort of a double edged sword- it's a compliment, but could bring extra stress as well. Our most beloved para has a daughter on my caseload. Thankfully she's always been really happy with everything that I've done, but I was definitely more nervous doing our initial referral meetings/initial IEP etc. with a coworker.
I also think choosing a teacher based on a specific style isn't necessarily an insult to the other teacher. That's why I partly disagree with our practices- sure, everyone is a good teacher, but there are different personalities and it makes sense to me to try to match those up. We have a very veteran 4th grade teacher who is famous for including a TON of drama/arts/acting stuff into her class- she even has a stage in her room! Some kids LOVE that type of stuff. Other kids would rather die. Matching personalities would help.