Should I bring derogatory remarks made on social media by a co worker to the attention of HR?

Write the link down on a piece of paper. When no one is looking, place the paper on the desk of an HR employee. Let them do with it as they deem fit.

I would not attach my name to that sort of a complaint.

I am confused by your grammar. 'They' and 'their' are plural, indicated more than one person. One person is 'he' or 'she'.

To answer your question about however many people are posting insulting things, I always mind my own business about what other workers do, and eventually a supervisor finds out anyway.

Ordinarily I'd say to leave things that happen away from the workplace out of it--but if he uses this account for work-related posts that clients or customers are likely to see, someone in a position of authority over him needs to know about it.

Your work environment should probably determine whether you first take it to his immediate supervisor or to HR.

The person has every right to call people what she wants regarding political posts. Those posts are entirely separate from any work-related posts. Just because she comments on both does not relate the two. She has a right to promote an upcoming event related to work. If the business does not want her to, it would be their position to tell her so. You can bring that to their attention if you don't think her work-related posts are appropriate but her political posts are nobody else's business and her employer has no place in injecting themselves in her political persuasion. If you are thinking of reporting her to the employer because you don't agree with her political views, think again. You might find yourself in hot water. It is not your place to decide that her political views are against the organization's values and even if they are, she has a right to her own political view - separate from the organization's view and a right to post her own opinions. IF she is posting on her own personal time, not on company time.

You can; they may regard you as a tattletale. And, they may let it go.

no

Dont follow the person if their comments offend you and tell them that. Social media is free speech. Why would you tattle on the person to your company?

If the business name shows up next to a questionable tweet, yup, bring it to the attention of your superiors.

HR would just quietly pass the information onto the executives in the company for them to make a decision on. This person will either get fired or get just quietly told to knock it off and then possibly get you labeled as a tattle tale. You know your company culture better than anyone here. It would depend on what your feelings are, but I personally would mind my own business or maybe just suggest that the company should only be tweeting from a corporate account to promote events if you had any influence over that.