Suppose an otherwise observant and all around excellent shochet is discovered to eat non-kosher food on a regular basis. (Just taking a random example, he's keeping the non-treibored hindquarters of land animals rather than selling them to non-Jews). Are we permitted to trust his shechita? Or does this knowledge disqualify him from service?
The Rambam in Hilchot Schechitah 4:14-15 appears relevant to your case
A Jew who is an apostate because of his transgression of a particular transgression who is an expert slaughterer may slaughter as an initial and preferred option. A Jew of acceptable repute must check the knife and afterwards give it to this apostate to slaughter with, for it can be presumed that he will not trouble himself to check [the knife].
If, by contrast, he was an apostate because of worship of false deities, one who violates the Sabbath in public, or a heretic who denies the Torah and [the prophecy of] Moses our teacher, as we explained in Hilchot Teshuvah, he is considered as a gentile and [an animal] he slaughters is a nevelah.
[Even though] a person is disqualified as a witness because of his violation of a Scriptural prohibition, he may [still] slaughter in private if he was an expert. For he would not leave something which is permitted and partake of something that is forbidden. This is a presumption that applies with regard to all Jews, even those who are wicked.
In the notes R Eliyahu Touger explains
Although he repeatedly violates that particular transgression, we do not assume that he will not slaughter correctly.
In his Kessef Mishneh, Rav Yosef Caro cites Chullin 4a which states that as long as if given a choice whether to eat kosher meat or non-kosher meat, the person would choose the kosher meat - even if he would partake of the non-kosher meat if kosher meat was not available - it is permitted to partake of an animal he slaughtered. The Kessef Mishneh continues, explaining that as long as one does not transgress with the intent of angering God, one may partake of an animal he slaughtered.
Although it also cites the Rambam's view, the Shulchan Aruch (Yoreh De'ah 2:6) mentions the opinion of the Tur and others who rule that if the person is not an apostate with regard to partaking of non-kosher meat, it is not even necessary to check his knife. He may slaughter in private. If, however, he is an apostate with regard to partaking of non-kosher meat, his knife must be checked. Moreover, if he shows no concern for kashrut at all, his slaughter is not acceptable (Rama Yoreh De'ah 2:5).
This is of course not meant to be applied in practice - and is only one of possibly many sources on the topic. See for instance SA YD 2:2 and ff