I'm assuming that when you say vertical pivot, you mean the pair that allow for vertical motion, rather than the ones oriented vertically - I used to think that identifying them by their orientation was right, but I was wrong. If you can rock them a couple of degrees, that seems pretty loose, and if you are handy at fine work, you can probably tighten them a bit, if you want. But you could also leave it alone - you won't damage anything, though you might be missing out on some detail in the sound produced
If you do want to tackle it, you need a flat head screw driver that fits properly in the screw (pivot pin) in the center of the bearing, and a spanner that fits the two slots in the locking ring. There are all sorts of kludges for the spanner, but most will mar the surface (as will even a correct spanner if used carelessly). An appropriate size flat head screwdriver with the center filed out so that you have two tines that fit the slots of the locking ring is excellent.
You only need to do one of the bearings. Back off the locking ring just enough to allow the screw (pin) to move (if you go too far, the pin may not stay in the bearing and then you'll have to locate the proper place for it, which is an added and unneccessary pain). Gently screw the pin in until it makes contact - gently, not hard, since you don't want to mash (brinel) the bearings. Back off an eigth of a turn to allow enough slop so that friction is minimized (there are bearing that work tight, but they cost the earth and aren't found (or needed) in commecial turntables). Holding the pin in place, and the spanner or cut out screwdriver (at an angle to clear the small screwdriver) turn the locking nut to mostly tighten it. Then tighten more fully, offering resistance to the pin turning, if you can. Finally, give a bit of torque with the spanner or cut out screwdriver to fully tighten it. The pin should not move much at all.
You can check your work by removing the stylus, balancing the arm, and seeing if it oscillates freely. Paul (Delihaus) used to suggest eight oscillations as perfectly fine, but this was with the arm held up and down - in normal position, the range of motion is so limited that you may get less. But it should oscillate several times - as a test, see how your arm does right now, so you have a standard of comparison that is fair.