You're generally not supposed to declare the stuff you already paid for and imported legally but you might be required to prove you did not buy them abroad, otherwise the regular import rules apply.
The details will depend on the specific country and a few other things (e.g. whether you are a resident, currently moving to the country, etc.) but you are typically allowed to import things worth between USD 500 to 1000, for your personal use, if you carry them with you. In all the counties I know a little about, this allowance cannot be shared between several people to import a more expensive item.
Anything over that value is typically going to be dutiable (there are very complex rules on how much duty you have to pay on what type of goods) and taxable (including VAT in countries that have it).
Now, the question becomes: What about the stuff you take with you on your trips? Between the Apple laptop from last year, the new iPhone, the designer clothes and your Swiss watch, you're already over USD 500. In practice, if it's all regular used tourist gear, customs officers do not bother. From the perspective of the country you are visiting, you will take everything back with you soon and aren't importing anything for good. From the perspective of the country you come from, it has probably all been properly imported and taxed before.
Being too picky about those things would mean expending considerable resources for a limited return and would put an unreasonable burden on either citizens or the tourists that come to spend money in the country so it seems reasonable to tolerate it. Because of this, if you throw your old watch or go to Switzerland without one, buy a new watch there, mail the receipt and come back with the watch on your wrist, you might just get away with it. But you are still supposed to comply with customs rules both in the countries you are visiting and the country you reside in and the burden of proof might really be on you.
If you want to do everything by the book and avoid paying duties on goods in transit or used during a short visit, you should secure some evidence that the goods have been properly imported (to be able to reenter the country you initially left), provide some guarantee that you will reexport them soon (to be able to enter the country you are visiting without paying the full importation costs) or follow some temporary exportation/importation procedures. For expensive equipment or perhaps a car, customs might just bother and I know people who had such paperwork for their gear. Among other things, that's what the ATA carnet and the carnet de passages en douane are for.
As a random example, here is what the rules in Canada look like:
Canadian residents may temporarily export personal effects for use on trips abroad.
On returning to Canada, it is the individual’ s responsibility to establish that such items were initially taken out of Canada and were not acquired abroad.
So formally it's really up to you to prove you imported the ring properly, even if you bought it a long time ago and wear it all the time. And jewellery appears to be especially sensitive:
Most jewellery items, with the exception of watches having serial numbers and original pieces of jewellery that are numbered by the manufacturer, are not uniquely identifiable. Since the Y38-1 label is unsuitable for use with items such as jewellery, individuals taking valuable pieces of jewellery or other similarly non-identifiable articles abroad should be aware that the CBSA will not document such goods on Form Y38. If an individual wishes to take steps to avoid unnecessary delays and facilitate reimportation of these articles, an appraisal report should be obtained from a qualified gemmologist, jeweller, or insurance appraiser, together with a signed and dated photograph of the jewellery. This should be accompanied by written certification that the jewellery in the photograph is the same jewellery identified in the appraisal reports. Individuals should be aware that this appraisal documentation may be expensive to obtain.
Jewellery is a sensitive commodity given special attention during CBSA clearance. Individuals who are unable or unwilling to obtain documentation should consider leaving such jewellery in Canada to avoid problems when they return.
Failing that, you could be asked to pay duties or even a penalty upon coming back to the country. The customs do not even have to prove you bought the ring abroad, you have to prove you did buy or import it in Canada earlier.