Torque in Gears

So neither of those statements are true in general. Here are the conditions that make each one true.

In the diagram there is a equal and opposite contact force $N$ between the two gears at the contact point. This force can be tangential to the gear, but in real life it is always pointing inwards. Also note the perpendicular distance (moment arm) of this force to each gear center.

The torque on A due to the contact is $tau_A = N r_A$, and the torque on B due to the contact is $tau_B = N r_B$.

$$ N = frac{tau_A}{r_A} = frac{tau_B}{r_B} $$

The only way this torque matches both sides is if the gears have the same moment arms $r_A=r_B$.

Now it is not necessary for the input torque $T$ to match with torque $tau_A$. Why? Because if the gear is accelerating then some of that torque is lost

$$ T = I_A alpha_A + tau_A $$ is the equation of motion of Gear A.

You can translate the above in terms of the tangent force $F_A = T r_A$ and the contact force $N = tau_A / r_A$

$$ F_A = frac{I_A alpha_A}{r_A} + N $$

Again the tangent force is only equal to the contact force if the gears are not accelerating.

Using Newton's 3rd we can see the contact force $N$ is also applied to Gear B.