"Sweet enough bribe" usually means paying cash over 99 turns, in the range of between 17 bct and 70 bct. Always cheaper to pay the stronger civ, because the price depends on relative military strength. You may even see the weak civ say, "No way, they are too strong."
Your diplomatic rating also matters, which is why a lot of Maso players seem to play with high Diplomacy picks a lot. Waiting until you build or buy Translators also helps.
I consider it exploitive to buy everyone to war with everyone else, but short of that, I regularly make use of a bribe here or there. Sometimes it is the only way to avoid becoming the attack target. Sometimes you need the help just to buy your civ time to get on your feet.
Rivals warring is always good for you if the rivals fight to a draw, as both are pouring resources down the tubes. However, you can undo yourself if you start a war and one of the sides actually beats down the other, as that can worsen your overall situation.
Good luck.
- Sirian