Should I address my professors by first name in emails if my peers are doing so already?

In the US-influenced culture of international Academica, closing an email with the first name can most of the time be interpreted as implicit invitation to interact, at least in further email communication, on a first-name basis.

So yes, most likely you can address your professors by first name in emails. Not because your peers are doing so, but because your professors are inviting you to. For other communication, I would just ask: "Is it OK if I call you Jane, or do you prefer Dr. Doe"?

At least in my field and in the US, the culture is that graduate students are supposed to be 'colleagues-in-training'. As such, it is natural for students to be on casual address terms with professors: they are supposed to be working towards being equals.

Of course there is still a gap in power and experience, but casual address is one way to start narrowing that. If others are using casual means of address, it's probably appropriate for you to do so as well.

Since this question is not tagged “US”, I'd add two data points from other countries that I know from experience:

• In Germany, the default is that professors and students, as well as other adults who interact in a professional relationship (except in performing arts), call each other by last name (not with academic title but generic “Sehr geehrte Frau Schmidt”, “Lieber Herr Müller” etc.). Oftentimes people switch to using first names, but this is an explicit decision and generally reciprocal – either both call each other by last name or both by first name (they are “per Du”). Many professors prefer the more informal first-name basis even for master students, and if the professor calls you by first name then it is probably safe to assume they're fine with you replying in the same way. If the professor is relatively young, then this would be probably the most common scenario; if they are substantially older then they would more likely explicitly ask whether the student is ok with first-name, or else stay with last-name (not completely uncommon even among colleagues).
  I believe this situation is similar in many other European countries.
• In Norway, and presumably the other Scandinavian countries too, virtually everybody calls everybody by first name. Even undergraduate students wouldn't think twice to address their professor “Hei Kjersti”.