Depending on state/county laws, you could be liable criminally or civilly. A "he said, she said" in criminal court means absolutely nothing without clear cut evidence and is mostly laughable with circumstantial proof.
On a civil front: If my chickens were killed tonight after reading this post and I decided to engage a lawyer who would take the case, I could sue you. Doesn't mean I would win, have a day in court, or ever even talk to a judge. But I could sure make you spend a LOT of money in replies, court orders, etc., before we ever got to see a judge. It's called "lawyering someone to death".
And if for some reason I was able to receive a judgement against you, I would still have to pay more money and convince another judge that I could actually collect $$ against you. This process is typically more involved in the ones above!
Point being: the term "Sue Me!" (in mocking) actually has a lot of merit since going after someone in a civil suit means virtually nothing unless you have a strong case and a large bank account to back you up.
Livestock is a completely different matter, but I have ended many conversations with (lawyer approved): "I appreciate your concern, but without some sort of actual proof of your claims you will need to retain legal counsel and consult my own". If you tell the accusatory neighbors the same, it will more than likely further irritate them, but make them understand that you know your rights and are willing to defend them. Engaging them past that only opens you to more animosity and conflict in my experience. However, focuses the lines (from their perspective) of what I think, and what I think I saw.
Sorry for being so vague, but just my 2c.. Worth as much as you paid.