https://www.livestrong.com/article/136464-fetal-ultrasound-risks/
https://www.newscientist.com/article/dn1639-fetuses-can-hear-ultrasound-examinations-.html
https://www.midwiferytoday.com/articles/ultrasoundrodgers.asp
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2390856/
https://www.inchem.org/documents/ehc/ehc/ehc22.htm#SubSectionNumber:6.3.2
https://www.midwiferytoday.com/articles/ultrasoundwagner.asp
https://www.asrt.org/content/News/IndustryNewsBriefs/Sono/studyshows062408.aspx
I do believe atleast 1 ultrasound during pregnancy is beneficial, preferably around 20 weeks. It should be no longer than 15-30 minutes of exposure. Ultrasounds are typically over used and can cause problems like thinking a baby is too big when they are actually 6 lbs (happens a lot actually). Manual palpatations, like MWs typically use, work just a good as a ultrasound for many of the things Doctors use u/s' for. Obviously palpatations can't pick up a heart defect in a baby, or another abnormality, but that is why 1 is a good idea around the 20 week mark when it is easy to see baby. Palpatation can also determine where the placenta is, as well as listening with a fetoscope. If a woman or baby has a medical complication that can be detected/watched/etc. with ultrasounds then that is a different story. But a normal low risk healthy mom and baby doesn't need more than 1 or 2 ultrasounds in a pregnancy.
Doppler radar is hard to find in terms of risk because it is fairly new. Doppler uses more sound waves, so if there are risks to a u/s then it would stand to reason there are risks to a doppler. Just because they haven't found risks doesn't mean there isn't any, and I'd rather be safe than sorry. For all I know is there is a large doppler study going on right now that'll be released in a few years. A Doctor should be capable of using a fetoscope. There is NO risk to a fetoscope at all, plus you can hear the actual heart beat rather than the sound waves like a doppler.
I am not comfortable advising mother's to use a at home doppler. Not enough research to prove they are safe, plus a doppler can give out false security and many women pick up the placenta rather than the baby, or their own heartbeat.