Is it common or something usual for students to sometimes go home for the summer?
It would be very unusual for an international PhD student to leave for more than a month to visit family, at least within the sciences. Students are paid during the summer. Leaving for the whole summer would likely entail giving up the pay, which many students cannot afford.
Some international students do not leave at all for financial reasons, or for fear they will not be able to re-enter the United States.
This is totally different from undergraduates, who usually leave the university during the summer.
I have seen a wide variety of decisions about the use of the summer break from classes, both in the case of students from the US and those from abroad. It really runs the gamut. Examples:
One friend couldn't afford the air fare to visit his home town, during his entire PhD. This was very stressful for him.
One friend made his good-byes in May and came back in late August married, with his wife accompanying him.
Some students visit family for part of the summer.
Me: when I needed the money I worked as a secretary in a temp agency over the summer. Otherwise I enjoyed the break from TA'ing, which allowed me to immerse myself in my exam prep or my project.
It's a very individual decision. I don't think one should decide about summer plans based on what other people do.
International students do visit their home country. I have seen several Ph.D. students spend their entire summer in their home country. However, it completely depends on the student. From my perspective, I think it is a bad idea to waste your three months by going back to your home country. Either you should do an internship or you should continue your research with your adviser. I explained the reasons in the following paragraph:
1) If you choose to do an internship at the industry, it will help you in various ways. The experience in the industry will help you to finalize your career goal. You will get the chance to experience the industry work culture, you will get a chance to work with some industry leaders and it will be a valuable experience for you during your degree. The experience eventually helps students to get a job after your degree in the industry. If the student performs well, they even get a return offer. I have seen multiple cases where the student landed a job after their internship. If the student decides to go to academia, even then the industry experience is valuable. Moreover, the internship helps a student significantly monetarily. Generally, the companies pay the same salaries of a full-time employee to an intern and the amount is pretty high for a graduate student.
2) If you are a PhD student and do not want to go to an internship, then it is better to work on your research with your adviser. Eventually, you have to finish your thesis, so it is better to utilize your time. Generally, the advisers have funds to support their students in summer. So, if you get paid then it is better to work on your research during summer. If you do not get paid, still it is better to research then going back home.
3) If you are a masters student and do not get an internship, then it is better to find some positions in research labs and work with a professor. Sometimes, professors look for masters students and hire them for small research. Any academic work can be added to your resume and eventually help you to get a job later.
If you are a masters student and do not get an internship/works to do in summer, then you can go back home to save some money.
I am a Canadian working on my PhD in the US. I was able to get a job in a clinic at home that does very similar research to my lab in the US and was therefore able to go home for June - mid August. I would say if you can do something like this where you stay productive over the summer, then yes it it okay to go home for the summer. But grad students do not typically get the summer off like in undergrad... grad school is a year-round job.