"By me" for אצלי isn't standard English, it's Yeshivish (an English sociolect.)
From Wikipedia: "The preposition 'by' has a wide array of meanings in Yeshivish […] A possible cause for this is that the Yiddish preposition 'bei' is defined as at, beside or by. The similar-sounding English preposition 'by' has come to encompass these meanings."
It's from before I was born, so a second-hand knowledge. One popular scholar, Ruvik's Rosental, claims so in http://www.ruvik.co.il/%D7%94%D7%98%D7%95%D7%A8-%D7%94%D7%A9%D7%91%D7%95%D7%A2%D7%99/2003/25042003.aspx; I think I read it elsewhere, too, as a common knowledge, but not 100% sure. I guess the signs for a French influence is (a) at the time (and still now), a French aura is more suitable to make a business look attractive. (b) At about the same time, many businesses of partners took the name "כהן את לוי". This את is obviously meant to mean "and" - but how comes, when in modern Hebrew it doesn't mean that, and actually sounds weird at first? Indeed it's half-justified by the fact that in Biblical Hebrew את meant "with", and it survived in modern Hebrew when inflected (איתו etc. from an earlier lesson in the course). But the real reason for coming up with this את, again according to scholars and maybe a common knowledge, is the french "et".
Correct solution: The shoes are at mine. ??? This can only make sense in English (which I speak natively) if: 1) אצלי means 'at my place' AND 'at mine' is elliptical for 'at my place' - in answer, say, to someone saying that the shoes were at his place. -OR- 2) אצלי means 'with me' in the sense of 'in my keeping'.
Native English speaker and fluent Hebrew speaker here. I was also confused, אצלי doesn't mean 'at mine'.. i don't even know what 'at mine' is supposed to mean. אצלי means something more along the lines of the 2nd definition you wrote - i have it in my keeping. There is a difference between איתי - which means 'with me' in a more basic sense, and אצלי - which means 'with me' in the sense that you have it with you. This difference is slightly hard for me to explain, but i'll do my best. If someone is at your house, you would say they are אצלי בבית - like @airelibre suggested. If someone is simply with you - as in walking with you, you would say they are איתי. One would never say הנעליים איתי, because shoes are never simply 'with someone', they are always with someone in their keeping. The same goes for other objects - המחברת אצלו (he has the notebook/the notebook is with him), אצל מי החולצה שלי? (who has my shirt/with whom is my shirt). I hope this helped you. Good luck!
Native English speaking American here, who has lived in AU and NZ over the past 7 years. I associate "at mine" with British English (which would include this region) as a term, as I wouldn't imagine an American saying "The ___ are at mine" or "Let's meet at mine", though I hear (and say) it a lot after living over this way.
Bob Dylan: "I'll Keep It with mine" https://vimeo.com/182493352 Doesn't entirely relate but it's one of my favorite Dylan songs. But regarding אצל: according to Giore Etzion (Routledge Introductory Course in Modern Hebrew, 192) אצל can mean "in one's possession" in addition to "at one's place." An example is הכרטיסים עצלך = יש לך הכרטיסים. Both mean "you have cards." Etzion writes that this prep. typically follows a verb that describes things that you do at a place such as ללון, לבקר, להיות. When there is no verb, it's only clear from context whether the prep. has the locative or possession sense. It seems that DL's translation, "the shoes are with me," combines location and possession.