I'm starting a grad school and I'm used to start my emails to all my professors with "Dear Professor Smith", even though they respond with their first name at the end of their emails ("Best, John").
I noticed that the guys I'm starting the studies with are using "Hi John" in their emails. I don't think they know professors better than I do.
I don't want to be too familiar, but neither be the only who keeps this distance. Should I switch to "Hi John"?
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.
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"?
Since this question is not tagged “US”, I'd add two data points from other countries that I know from experience:
Signing an email with their first name doesn't immediately grant permission to address them in that way. Specially in academia. Ask them how they want to be addressed.
Let me add an anecdote. I used to work with a professor whose personality often resulted in people talking to them by their first name. However, I always referred to them as "Professor", both in person and via email. They never told me to do otherwise and I was completely fine with this.
A couple of years passed and one day I decided to ask if they'd be OK if I referred to them by their first name. This resulted in a whole conversation where they expressed how upsetting it is for people to take this for granted (especially undergrads). The conversation ended with them saying that of course it was fine and thanking me for asking.
FWIW this was in North America with an United-Statesian professor.
Different places have different standards, even within the same country. The US tends to be fairly informal, but I was the only grad student who generally called the professors by their first names. But I was a bit older than most and the faculty was young, so there was little social separation. On the other hand, I pretty much required my own doctoral students to use my first name, though some of them found it hard, as they came from more formal societies.
My preference, personally, would be to stay fairly formal for a bit until you at least meet them face to face and take our cues from that meeting. You can't really go wrong by being a bit more formal than you need to be.
But, in the US, at least, and with your description, you are probably safe enough.
Simple rule of the thumb is: address anyone the way he wants to be addressed. Some prefer prof...others prefer Mr/Mrs/Miss. Find out from them how their correspondence should be addressed.
In Nordic countries and in mathematics, the norm is to not use honorary titles in general. Hence, I would be starting my emails with "Hei/Hej [first name]", in general. Foreigners coming from hierarchical cultures might not be used to this, so if writing in English, I would typically write "Dear [title]," and thereafter copy what they do.
Of course it depends highly of where you are located. There are some countries and universities in which there is a strict social hierarchy and professors prefer to be addressed by their title (some conservatives regions in Asia come tome mind), and there are others where the boundary professor-student is rather blur (where I live professors would feel awkward if being addressed as Dr., given they speak casually with students, share drinks with them while lecturing and so on).
I would consider three points:
When I was a grad student in Canada (in engineering) back in the 70s/80s, students always addressed professors as either Dr. Foo or Prof. Bar, never by just their first or last names. I can't even imagine someone doing otherwise.
I was in a professional Masters program 5 years ago in the US (in software engineering). I was older than perhaps two of my profs. I believe my entire class (which ranged in age from perhaps 30 to me) addressed all of our professors using doctor or professor honorifics. Now, one of them was Prof. FirstName LongName-AnotherLongOne, I called her Prof. AnotherLongOne.
