There is no way to know the actually supply and demand in the neighborhood, or the urgency of the seller to sell--- those are factors you can't include in an appraisal.
Appraisers get as close as they can to guesstimating the value based on sales in the past 6 months within 1/2 mile.......without knowing where the chips will fall based on today's buyers and the sellers needs.
Start low...you can always increase your offer later, (as long as the starting offer wasn't too low where they might be insulted and not want to hear from you again.)
Your agent should be directing you with the most recent comps sold as well as competing homes. If your offer is cash, or has other great aspects like flexible closing, inspection forgiveness, willingness to get certs etc . this could bring the price down. There often is a cash discount though it depends upon motivation and needs of seller.
I would recommend that though sellers may have done appraisal you still have an appraisal clause in your contract for added protection.
The appraisal is by no means the answer to the pricing of the home.. yes of course buyers want the house to sell at the appraisal price level and many fight to try and do so. I have seen an average of about 85% of appraisals that are incorrect to the market because the appraisers do what the bank wants them to do numerically. Why? To continue getting business! ( that is from a Trulia Realtor/appraiser)
You should have your Realtor put together a CMA for the property with sales in the area, pricing history an analysis of the decline in the particular area you are in then formulate an opinion on the price.
Remember, Real Estate is local.. Appraisers are picked from a pool.. so you may get a guy from Hackettstown driving to Monmouth county with no idea of the area sales, comps nothing... and you get a bad appraisal.
Due diligence from all angles will help you make the best pricing decision.
The start low comment below is usually the worst thing you can do. You will insult the sell and then, (from experience) I find that my sellers, and sellers in general will totally ignore a lowball offer and then look to get more from the "lowball Offer" buyer than they would normally accept for the house.
The appraisal number is not derived from the comps alone, but the appraisers value of what is in and around the house.. if it were just comps a Realtor BMA would be just fine...
Again, due diligence from all angles will help you make the best pricing decision.