Using p-channel mosfet on high side of h-bridge, do I really need a driver chip?
Firstly, I think you have your P ch and N ch mosfet symbols reversed. Secondly, no, you don't need to use a driver chip if the top mosfet is P channel (providing you are not making extensive use of PWM to control the motor).

If you are using PWM then I'd suggest you use push-pull drivers because the gate-source capacitance on MOSFETs is usually in the order of 1nF to 10nF and "charging" this amount of capacitance from a GPIO pin takes several micro-seconds. Worst still, the resistor that discharges the gate will take significantly longer if the resistor is in the middle kohm range.

So, if you are using PWM then I'd go the whole hog and use a driver AND use both N ch devices - efficiency will be a tad higher than using a P channel device as the top FET.

As an example use the following formula: -

\$\dfrac{dQ}{dT} = C\dfrac{dv}{dt}\$ = I (current injected into the gate capacitance)

So if capacitance is 3nF and you can inject 1 amp, the voltage rise on the gate is 333 volts per microsecond - you'd be looking for a rise time less than 1us to achieve maybe 10 volts and that sounds to me like a rise time or fall time of about 30 nano seconds.

If you were relying on a 1kohm resistor to discharge the gate, CR time is 3 microseconds and in reality you might need about 10 micro seconds to discharge it properly.

The option is yours.