I respond to every offer I receive. If I find the offer really low, I will just say "No, thanks for your interest." If they are really interested, they will make a better offer. Negotiation can start from there. I have made more than one sale that way. If also helps to weed out people not interested in negotiating.
On the other hand, I had one item for sale. To make the example easy, I am going to say the asking price was $1000. This item routinely sells for $1200 and more. The potential buyer and I haggled back and forth a bit and settled on a price of $850 with shipping split. I made arrangement for this person to pay and waited for some time and got nothing. I contacted the person and was asked if I would take $600 for the item. I told him we already had an agreed upon price. Crickets for a couple of more weeks and then he asks if I would sell for $500. I responded with, "For you, I will sell for $950 and you cover the shipping." Never heard from him again. The item ended up selling for very close to my original asking price, so it ended up working out fine for me at least.