The nature of Overwatch being so complex with different abilities/ults for each hero and varied comps, it would be hard for one person to keep tabs on everything going on and relay the information to his team.
I have never personally been in a professional setting so this is my take on how the general calling structure would be.
First would be ult checking (allies) and tracking (enemies), which historically would fall under the main support role (so the Lucio, and recently Mercy player). Calling for an ult check helps everyone in the team know where they stand in ult economy, preparing for combos, getting position if need be.
Tracking enemies ults allows for better preparation going into fights, expecting combos and getting ready for a team wipe (if dry fight) or a disengage+reengage (Fusion and XL are very good at this) for example. This is a very tricky one and in my opinion makes or breaks a main support at a professional level.
- Supports ults are usually mirrored, but you can see exceptions here and there (i.e. winning a team fight and getting to heal everyone as lucio, but in the long run it normalises across teams).
- Tanks' damage is pretty easy to keep an eye on, them being on the frontline, I'd say Roadhog would be the toughest to keep track of, but Wholehog is hardly impactful so not a big deal.
- Trickiest one would be the DPS ults and their usage imo.
Second is scouting. In most cases made by the Tracer player. This is for getting a better image of the enemy's position before engaging. The better you are prepared for a fight, the better are your chances at winning it.
I'd imagine Sombra players would easily be able to do this as well, but she's not as omnipresent as Tracer has been in the game. Also high ground allows for Genji, Widow, Winston and Dva to do the same in specific situations (Volskaya, Anubis, Hanamura 2nd, King's Row 3rd etc).
Then is target dive/focus. This is Winston's job in the dive meta. Where to go and who to go on. Bubble is a great ability to spread a team apart and for the DPS and Dva to deal damage 'safely' for a short amount of time. There is also the issue of Winston not having any range so you need him to be the one making the call. Seeing the good leap, bubble usage and decision making are massive parts of Winston's no brain no aim job.
- Last point would englobe every small thing that happens in the midst of fights (no reflect, no recall, low matrix/shield, hook's up, Ana/Zen dropped from x position, peeling etc), that would help make engages more calculated and effective. And this is easier for frontliners to spot and call. I'd call it counterplay.
Also keep in mind that calling affects mechanical play as it takes focus away, which is why DPS players don't call as much in general.
So all in all, I'd say everyone is involved in calls. Some less than others depending on the team's calling structure.