My notes are below. For players who have played within the past 15 years, I have listed this from my own viewing experience. I haven’t followed every team closely so others may have something to add. For older players, if they were a RHS and were normally partnered with LHS I assume they played RD primarily. If they were a LHS and were normally partnered with a RHS I assume they played LD primarily. There were also several seasons in the 1930s when all-star voting was separated by LD and RD, which can provide useful information.
Orr – He’s played RD in any games and highlights I've seen. Partners were always LHS. Not aware that he ever played LD in the NHL.
Harvey - I've read that he and Tom Johnson were the two RD for Montreal in the dynasty years and seen highlights from those years of him on the right side. As a younger player he was paired with Butch Bouchard, a RHS - I assume he played LD while playing with Bouchard.
Bourque – I remember him mostly playing RD in Boston, but he played LD in Colorado while paired with Foote or Blake.
Shore - RHS, most of his all-star votes came for RD.
Lidström - Played LD for the latter half of his career that I saw. All his partners that I'm aware of were RD - Coffey, Murphy, Rafalski, etc.
Potvin - He played a lot with Ken Morrow, a RHS - so I would guess he played at least some LD but don't know.
Fetisov - In the 1987 Canada Cup I'm pretty sure he played the left and Kasatonov played the right. Would like to get more information on the rest of his career.
Park - Played RD for Canada in 1972 - didn't play with Orr at even strength because they both played RD.
"King" Clancy - Based on all-star voting - LD in 1930-31, unclear in 1932-33, RD in 1933-34,
Coffey - Played RD in Edmonton for sure, not sure about all his other stops.
Pilote - Partnered with another LHS, Elmer Vasko. Not sure which one played the right side.
Seibert - Received all-star votes at both LD and RD - BUT his LD votes came in years the league had many high-end RHS defensemen (Shore, Clapper, Coulter, Seibert).
Pronger - In St Louis, Anaheim, and Philly played primarily RD. Played LD in Edmonton (with Jason Smith) and for Canada in international tournaments.
Leetch - I assume he played LD while paired with Beukeboom?
"Dit" Clapper - Received all-star votes at both LD and RD - BUT his LD votes came in years the league had many high-end RHS defensemen (Shore, Clapper, Coulter, Seibert). His most frequent partner (Johnny Crawford) was a LHS.
Savard - Famous for his Savardian spin-orama, which he used while playing on the right side of the ice.
Niedermayer - Played RD in New Jersey as far as I remember.
Conacher - Most of his all-star votes came at LD
"Ching" Johnson - Most of his all-star votes came at LD
Chára - Played mostly on the right side in Ottawa with Redden, then Phillips. Has played mostly on the left side at ES with Boston.
Goodfellow - Most of his all-star votes came at LD
Bouchard - I assume he played RD - RHS.
Brewer - Played LD while paired with Bobby Baun
"Babe" Siebert - Most of his all-star votes came at LD
Johnson - He and Doug Harvey played most of the minutes at RD for the dynasty Canadiens
Coulter - Received all-star votes at both LD and RD - BUT his LD votes came in years the league had many high-end RHS defensemen (Shore, Clapper, Coulter, Seibert). His most frequent partner (Muzz Patrick) was a LHS.
Wilson - Played a lot with Bob Murray, RHS.
Mantha - I assume he played RD - RHS.
Cameron - I assume he played RD - RHS.
Stanley - Played LD while paired with Tim Horton.
Edit: Adding a quote from Larry Robinson's book, which matches what Canadiens1958 posted below.
The mentorship started back in Montreal, when Serge used to pair me with every new kid that came to the team as a defenceman. I wouldn't say that it hurt my career, but a lot of times I was playing on the right side, where I didn't feel as comfortable. I preferred the left side, and that's where I played when I was on the blueline with Serge. But all of a sudden, I'd be playing with someone like Gaston Gingras, and because he played left, I was moved over to the right side (being more able to do that). That meant that a lot of times, when the puck came around the boards, I would be on my backhand. I had to reinvent my game and learn to play from the right side.