As far as I understand, the Bell test experiments show that observations (measurements) of entangled particles have an effect on one another.
Not quite. The Bell test experiments show that measurements of entangled particles are correlated with one another in a way that is impossible using either classical mechanics or hidden-variable theories. Correlation doesn't imply that an interaction is taking place between them at the time of measurement. It doesn't even imply this in the classical world: for example, suppose you and your friend divide up 50 pennies randomly and then fly to opposite ends of the country. Once you're separated, you count your pennies and note that you have 32 of them; this, of course, means that your friend has 18, but it doesn't imply that your pennies interacted with your friend's pennies to produce the split when you were counting them. The information was already there when you divided up the coins. In quantum mechanics, this still holds true, in a sense; though the results of a measurement aren't determined until it is made, the correlations in the system (i.e. the usable information in an entangled system) were already set when the entangled system was created.
This effect is apparent when examining the results of both particles, but what about examining only the results of one of the particles? Shouldn't they change their behavior -- say, their distribution -- when making a measurement on the other particle?
The thing that distinguishes an entangled system from a non-entangled one is correlations. You can't observe any correlation when you measure only one particle, and there is no interaction going on between the two components of the system when one is measured, so you can't tell when the other is measured just by measuring one of them.
To demonstrate with an example: suppose we have the classic entangled system: two electrons, prepared such that their spins are anticorrelated (when one is measured to be spin-up, the other is measured to be spin-down, and vice versa). Neither individual spin is determined until measurements are made, but their correlation is always determined. Suppose you make a measurement on your half of the entangled system, and you receive a spin-up result. Are you getting this result because the other person already measured their electron to be spin-down, or are you getting this result because you're the first one to measure their electron and it was randomly-determined to be spin-up? Without contacting the other person, there is no possible way to tell the difference between you measuring first and you measuring last (after all, even if the other person measures first, they also have no control over what the spin of the electron will be measured to be, so they can't send information by "purposely measuring the electron to be spin-down").
The only way this effect would not be apparent when examining only one particle is if the distribution of outcomes (either up / down) is 50-50 for both particles (no matter the axis).
While that's certainly the most common system that's brought up as an example, the probability of measuring an individual component's state really doesn't matter, as long as the outcome is uncertain. For example, suppose we prepare the system such that your electron has a 99% chance of being spin-up and a 1% chance of being spin-down, whereas your friend's electron has a 1% chance of being spin-up and a 99% chance of being spin-down. If you measure your electron to be spin-up, it's still impossible to tell whether a) you measured first, and your electron's spin was randomly determined to be up, or b) your friend had already measured their electron's spin and found that it was randomly determined to be down. Both processes produce the exact same outcome on your end, because neither of you have any control over the results of the measurement.
Is that the case, and is that why this effect is only apparent when examining both particles but not when only examining one?
Nope, it's all about correlations, as the previous paragraphs imply.