We performed this experiment and I am not really sure what to write down. I believe the pulley moves due to the static friction due to the string, so you need friction in order to perform this experiment. But the friction also requires additional tension force for the ropes to lift the string. What are your thoughts?
The point of this experiment is to show that the vector sum of the tensions in the strings which meet at origin O is zero. You do this by calculating the tensions $T_1=m_1g$, $T_2=m_2g$ and $T_3=m_3g$ in the strings and measuring the angles between the strings at O, and drawing a Triangle of Forces, using 3 arrows with lengths proportional to the forces and the angles as measured. If the triangle 'closes' this shows that the 3 forces are in balance. If the triangle does not close, this shows that some other forces (such as friction) are involved. (You can use more than 3 forces, in which case you should get a closed Polygon of Forces.)
Friction is not needed for this experiment to work; it is actually the main source of error. Ideal pulleys are equivalent to frictionless pegs or cylinders, around which the string slides freely. If the string slides over the pulley that is not a source of 'error'. If there is friction between the string and the pulley, so that the string does not slide around the pulley but the pulley rotates, there should be no friction between the pulley and the axle/pivot.
For an ideal pulley/peg the tension in the string is the same on both sides. Real pulleys/pegs have some friction, with the consequence that the tensions on each side of the pulley/peg can be different. See the Capstan Equation or Frictional force on a rope wrapped around a drum.
The consequence for your experiment is that, if there is friction at the pulley axle, then the tension $T_1$ in the string from which mass $m_1$ is hanging will be slightly different from the tension $T_1'$ in the string on the other side of the pulley. This is important because it is the forces $T_1'$, $T_2'$ and $T_3'$ in the sections of string which meet at the origin O which should be used in your Triangle of Forces, rather than the tensions $T_1$, $T_2$ and $T_3$ which support the masses $m_1$, $m_2$ and $m_3$. Your experimental method assumes there is no friction at the pulley axles, so these two tensions are the same - ie $T_1=T_1'$.
One way round the problem of the unknown friction in the pulleys is to place force-meters between the pulleys and the origin O, to measure the correct tensions $T_1'$ etc meeting at O. However, school laboratory force-meters are not usually very accurate, whereas the masses $m_1$ etc can be measured accurately on a chemical balance.
The alternative is to make the pulleys as frictionless as possible, or replace them with polished steel rods and test various types of string to find one with very little friction. With the same combination of masses you should repeat measurements by placing the origin O in various initial positions before releasing it and allowing the system to reach equilibrium, jiggling each mass in turn to check if the system will move to a 'better' final equilibrium position. Place a pin-prick at each equilibrium position O. Then choose a 'best fit' position of O which is at the centre of the pin pricks, and measure the angles for this position. Don't bother with the other positions.
In the 2nd diagram a 3rd pulley is used instead of hanging $m_2$ directly from O. One reason for this might be to obtain better accuracy by ensuring that there is friction in all 3 directions and hope that the friction forces cancel out - which might be expected if the 3 masses and the 3 angles are approximately equal. However, static friction can act in both directions in which each string can move, and it is difficult to tell which unless you check carefully by allowing the masses to move. A small adjustment can reverse the direction. The directions can differ between pulleys - towards O in one case, away from O in another.
You could 'quantify' the error ('uncertainty' is a better word) in your experiment as the gap-distance by which your triangle does not close, expressed as a percentage of the sum of the lengths of the 3 sides. Depending on your apparatus and the precautions you took, this error could be quite large. Using this measure, although performed quite carefully the error in this experiment was about 10%.
After friction, the 2nd largest source of error is measuring the angles between the strings. This error can be minimised by using a large apparatus, marking positions with pin-pricks, and measuring angles with a large protractor.
The pulleys don't need to move for this experiment. If we take ideal pulleys (that is with zero friction) then the extra tension due to friction would disappear and we would get the most accurate result. Thus if there is friction it would surely affect our accuracy but for ideal case, the results would be accurate.
In ideal case, the string would just slip on the pulley without rotating it.
The pulleys are assumed to be free to rotate about their axes (without friction in their bearings). The static friction between the strings and pulley surfaces assures that the strings won't slip relative to the pulley surface. Under these circumstances, if you perform a moment balance on a pulley about its axis, you find that, for rotational equilibrium to exist, the tension of the string on one side of the pulley must be equal to the tension on the other side of the pulley.