As a parent and ex-HT I can see both sides. It can work really well, or can be a real problem depending on many factors such as teachers involved, % of staff that are part time, size of school and flexibility of all staff.
My current large school has more job-share teachers than fulltime teachers and it does cause problems for us. Our issue is that we have many UPS3 part time teachers across the school, but because they are part time they refuse to take on much in the way of additional responsibilities. "I can't be maths coordinator, I'm only part time", "I can't run the eco-schools project I'm only in two days a week", "I'm only in three days a week, so I won't run a club", etc. So everything falls to the few fulltime teachers. This equals a huge additional burden for us, many of whom are also working parents and on lower salaries. Classroom teaching and sharing the basic planning and teaching is fine (for most of them!), but the whole school stuff is what is putting off our HT from maintaining any job-shares that have naturally ended, or in introducing more. The UPS issue cripples us financially from giving TLRs for other people to undertake these roles, and as the UPS is not for middle management type responsibilities the unionist say we can't insist the part time staff take on these responsibilities.
The flexibility (or lack of) of our part time staff to cover for absence or change their days is not a bonus for us or the children, as the vast majority can't or don't want to work on their non-contracted days.
Part time staff only attend staff meetings on a pro rata system and if the meeting is on their working days, so we have to have a complicated timetable of staff meetings rotating around three different days of the week, which impacts the after school clubs we can run. Parents want, for example, the gymnastic club on the same day every week rather than a rotating programme or alternate weeks that fits in with our staff meeting timetable. The attending teacher should take notes and feed back staff meeting outcomes to their jobshare, but this is always a subjective feedback and issues get missed, causing problems especially in important policy changes or training to use new systems.
Also, I've been looking at the PPA timetable; this is a logistical nightmare with so many part-time teachers.
Teachers can request flexible working hours, but they cannot demand them as the headteacher and governors have a right to weigh up whether it is in the best interests of the school and the children. They have to consider the request, so it always worth putting it in writing.