It's one of those things where there's no definitive answer. Most people think that scans/dopplers have been proven to be safe, but the truth is that they have never been proven to harm the baby - this means that while there is no proof that they harm the baby, there is also no proof that they are 100% safe either. As with most things in pregnancy and parenting, it's a judgement call.
Personally, I like to keep things to a minimum just in case, so I have the two scans and the MW listens to baby at anti-natal appointments, but that's it. As I said though, that's just me.