For the source text click/tap here: Chullin 3
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"All may slaughter and their slaughter is valid" was explained by Rav Ashi as referring to a non-observant Jew. If he eats non-kosher meat to satisfy his appetite or to save money, but will eat kosher meat when it is available, then if he is handed a sharpened knife that was properly examined and found free of nicks, he will presumably not squander the opportunity to make the meat kosher.
However, if he does not have an examined knife, then he cannot be trusted, for he will not bother to sharpen it. If he went ahead and slaughtered with his knife - which he should not have done - one can examine the knife afterwards, and if the knife was found to be fine, one can eat from his shechitah.
Ravina said that "all" refers to those who take upon themselves to do shechitah, because most of them know the laws - provided that they already did shechitah a few times without fainting.
We explore how non sacrificial slaughter in Devorim allowed for meat consumption.