I have attached two pictures. Which one is the right way to put the gate resistors? Are they the same?

 

The typical/conventional way is the one on the right.
Tahmid's blog: Using the high-low side driver IR2110 - explanation and plenty of example circuits
Max.

 

I have attached two pictures. Which one is the right way to put the gate resistors? Are they the same?

I'd put the gate bleed resistor directly from gate to GND - the other way is unlikely to cause problems unless you're "pushing the envelope".

The series resistor feeding the gate usually isn't required, in linear applications you could end up with parasitic oscillations - adding that resistor might be the thing to cure it. In such a case it would be called a; "stopper resistor".

If you were building a high efficiency SMPSU; the series gate resistor would combine with the gate capacitance to make a C/R time delay - the MOSFET would spend more time handling both V & A between switched states and get hot.