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Does everyone agree that when you purchase a diamond from over the
internet you should have it appraised? What about if it is from one
of the “trusted dealers” that contributes to this site.
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It depends on your purpose. If you're going to have the ring insured, you're going to need an appraisal anyway, so you might as well get it right up front while you're making your decision.
In my opinion you're safe with the analysis and critique provided by those "trusted dealers" you mentioned. From what I've seen, these guys are straight shooters with awesome merchandise at great prices. It would be pretty hard to go wrong with most of the popular names you hear bandied about all the time on this site.
There's another side to the coin as well though. Often a vendor will feel better if you get an unbiased second opinion. That way they know you're not just relying on what they say, and will have more confidence and little or no second thoughts about the purchase.
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How should I go about the appraisal? Should I have the stone sent to
someone in or out of the setting? Or should I send the stone by
itself, have it sent back to the dealer, and then have it set in the
chosen setting?
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It's much easier for the appraiser, and you get a more accurate report, if he examines the stone loose. Your second option is probably the better of the two, although a good appraiser can usually work around a mounted stone if necessary.