Coming from a current student and grower-upper of this new era of information, once three letter agency employee, who has done very personal tutoring, coaching, etc- I strictly keep a social media account (a pseudonym so to speak) separate from my real life identity. I hardly use social media with my own name. It may come from a lack of written qualifications yet (working on it) and my rather contrarian writing, but social media is not a good place to use your real identity unless it is strictly for friends and family unrelated to work or school.
I will often, even just in the process of citing works, look for the author's social media accounts. I like to get an idea of whether I feel I have, if I disagree with them or find their work too easily criticized, tinted them poorly when perhaps they have changed their mind, seem to be in a different state of mind now, etc. Be aware that if not protected (which I will also explain below) they can find everything you have ever posted, done, or liked. There's no guarantee even if protected they can't. Twitter is not incredibly safe.
Protecting your tweets on Twitter gives an air of elitism, and I mostly regard those folks as having something to hide (if you don't want to be seen, why bother with the website?), whether it be opinion or just not realizing the point of Twitter is some measure of social exchange. I agree fully with protecting your messages from anyone.
Following this, I would keep one for my real-life identity (or my expressed published name/identity) that is strictly for professional use. Example: speaking with another professional colleague, making a declarative statement on a subject and needing my qualifications available, announcing papers I release, announcing awards, generally just a portable resume and new-age linkedin since linkedin will become a joke as soon as people realize it's mostly falling out of favor with the newest generation of workers. All I get are scammers.
Students may add this account, but this would be the account where there will be no interaction with them whatsoever. Sneak in likes if you would want, but remember: everything you do that is against the grain they expect, whether it be liking a snarky contrarian comment, will be taken much further than you predict it to. The passionate kid will take the like as confirmation he's right, when he should likely be challenged and helped in making his argument/belief stronger. An exception would be a graduate after the fact whom I may want to promote in some way (assuming I have such clout if it were a professional account).
I would protect the DMs of absolutely from everyone, and give a contact email for questions that is officiated (@overpriceduniversity.edu) (maybe even a separate one), since a lot tends to get filtered from non .edu emails if they were to contact you (not always true ofc). There is certainly a good deal of information exchange happening on social media, aside from the social part- much of it is nonsense, but to see and understand the honest, kneejerk reactions of individuals to certain topics can help one understand the general public opinion, even if the topic is fabricated.
Ran across this site and found it interesting since it's like an actually good version of Quora which I despise. I hope my perspective helps a bit, since in these coming decades, this integration of life, the internet, schools online, and our coming pandemic of total information access obfuscated by FUD is going to be much more difficult to navigate for those who did not grow up in it.
In summation: my line is in bold but italic. No underline. /