This study examined whether sleep-position affected still-birth risk: Going to sleep in the supine position is a modifiable risk factor for late pregnancy stillbirth; Findings from the New Zealand multicentre stillbirth case-control study (2017-06-13)
They found that sleeping supine (on one's back) was associated with a much higher risk of late stillbirths:
Supine going-to-sleep position is associated with a 3.7 fold increase in overall late stillbirth risk, independent of other common risk factors.
However, when it came to left versus right, they couldn't draw a conclusion:
In the current study, the risk of late stillbirth was not elevated in women who reported going-to-sleep on their right on the last night and the last week; whereas in our previous study there was a borderline increase in risk with right sided compared with left sided going-to-sleep position on the last night. Further evidence is required to confirm whether right sided going-to-sleep position is associated with similar risk of late stillbirth to left sided going-to-sleep position.
They also looked at previous studies on the physical effects on blood-flow in the different positions.
As discussed in the comments, this paper's results are not strongly powered - they are suggestive, but far from conclusive.