It shouldn't be "I earned it so it's my kill. If you secure it it's a steal because I did 99% of the work" because kills aren't just to say "hey I have this many". They give gold, something your entire team needs. This is a team game.
They give gold. Some people may need the gold more than you do. Sometimes I end up letting an enemy go with 1% health (from full) because I know an ally can secure it, and they're a little behind on their build.
Quoted simultaneously for truth and stupidity. The point of a KS (kill secure) is to prevent unexpected escapes. Suddenly stopping and not finishing an enemy is a great way to make me hate you forever when I break off at the same time for safety because I know you can get it.
It's pretty simple really, don't waste mana or abilities you are likely to need later, and beyond that, everyone should endeavor to make as many enemies as possible dead as soon as can physically be done. A "ks" is only possible because that last bit of damage cannot be managed by the primary combatant(s) before the "stealer" can do it, so necessarily the enemy is dying sooner; this is a net benefit to the team, and if you are not "stealing" my kill when you can, you are a bad teammate. It only becomes a problem when people who can viably be helping the whole time are not helping and stand idly by until the last second to go for a kill - in other words, it becomes a problem when sustains play like Twitch. But Twitch playing like Twitch is not a problem, and neither is someone who came as fast as they could whomping a finisher on them with their opening move. Indeed, virtually all KS claims I witness are nonsense.
edit: oh wait a minute, I actually wrote a list of 9 things every LoL player should know... didn't post it anywhere yet

but that's on the list... let me copypasta...
5. There is basically no such thing as kill-stealing.
Win or lose, there is no reward for being the best performer on the team; only for being on a team that wins. When an enemy is killed, your team's total XP and gold are just as good regardless of who gets the last hit; in fact, the total gold for the team goes UP by 70% if an un-assisted kill becomes a "stolen" kill, because a seperate gold bonus is created and split up for assisters. If an enemy is almost definitely going to die, your team should make them go from "almost definitely dead" to "dead" as quickly and reliably as reasonably possible. A professional never hesitates and should never have to apologize for making sure that an enemy really really definitely died for sure. As a potential stealer, you are immensely overconfident if you think you can model all possible ass-pull escapes and rescues that the entire enemy team can pull off in all possible universes in the split-second between now and the ally's "rightful" kill; just secure the kill ASAP for the good of the team. The only exception is when a player has a soul-stacking item (Mejai's, Occult, Leviathan) and you don't, in which case you should try to feed them the kill if you can safely do so. Otherwise, there is no such thing as kill-stealing.