With proper management bougies can grow quite a bit in one season- question is whether or not this situation gives them what they need to do so.
Is there enough light there?
Will you be repotting them into larger pots with decent media and watering consistently?
Can they overwinter outdoors there?

Yes they get all the afternoon sun as the patio faces the west. I will also take them indoors in the winter... I just want my patio to be sheltered during the warm season because that's when I also want to get out on the patio and I want some privacy! I heard that you should not repot your Bougainvilleas unless you are really forced to do so because the plants require to be watered too frequently. That is because I heard they not only bloom better when they are root-bound, they also do not want to have their roots disturbed. They have plenty of soil in their pots and they don't give any sign of wilting at all so far... Instead they are making new blooms as I speak! Here is the patio from outside (The plants hanging are Passion Flower "Amethyst"):

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With any potted plant in the sun, I would recommend changing to a lighter colored pot, or using cache pots (with drain holes). Black pots are root-roasters! The thing about any plant performing better when it's root-bound is an old wives' tale. If such conditions were to cause an unusually heavy flush of blooms, which I doubt, it would be because it's the last valiant act of a dying plant, trying to survive via offspring.

Those passion vines look great, btw!!