For many teachers, there is an imaginary line about 5 feet in front of the chalkboard, near the table. Students do not cross that line, nor do teachers. One of the best Classroom management skills you can develop is breaking The Plane, or getting in the habit of moving about the classroom as you teach.
There's a great discussion of this in the book Teach Like a Champion by Doug Lemov. It's Technique 15 "Circulate". He suggests that teachers cross that imaginary line during the first five minutes of every class. He suggests that teachers must suggest to their students that they own the room, and it's normal for the teacher to move around anywhere in the classroom. I have found this to be very helpful in my class. You already know that approaching a chatting student can help redirect them. You may need to tap them on the shoulder and point where they should be looking, or you might bend down and talk. Practice moving around the room as you teach so that students become comfortable with the idea of you moving around. It will help you address problems quickly and with minimal disruption.
I have been trying to learn this technique myself. It is HARD to cross that line for the first few times -- it's like a force field! But now, my students don't think twice about me moving in and among their tables or looking over their shoulders, listening in on a group discussion, or answering a private question. What's more, it's much easier for me to move toward a student and cut off their talking without breaking the discussion or causing a big disruption. Most of the class isn't even aware of me stopping a situation before it gets out of hand, because they are already accustomed to me moving around.