conspiratemus1

I don't know for Italy, except that I presume that those itinerant markets that set up in small towns on market day and deal only in cash are not remitting VAT to the government. The stalls don't often even have cash registers.

VAT regimes typically have a long compliance code that describes in excruciating detail what is taxed and what isn't, flowing from the political realities of setting up the tax peculiar to each country. The Canadian VAT (we call it a Goods and Services Tax, aka the Gouging and Screwing Tax) is broadly modeled on British and European systems (with the additional complexity of our provincial sales taxes stuck in there as well.) So, based on Canadian practice, there are three ways that your used bike from a private seller (i.e., not an organized bicycle business) could be tax-free:

1) The seller is exempt because his/her total annual sales are less than a certain threshold and you thus avoid making the little kid running a lemonade stand register as a tax collector for the government. This would apply to nearly all private used-bike sales in Canada. But if you were running a used-bike business, you would definitely have to collect (and remit) VAT on the sales of those used bikes if the volume was enough to make it a viable business. Used-car lots pay VAT.

2) The specific good or service is exempt. e.g., new houses are taxed in Canada but re-sales aren't. Legal fees are taxable in all cases but insured medical fees are not. Rent and loan interest aren't taxed but utilities are. Famously (to Canadians) a single donut in a bag (snack) is taxable but a dozen donuts in a box (basic groceries) is not. Governments don't like to make too many things tax-exempt because it creates precedents where everyone else wants their gig to be exempt, too. It would be unlikely that sales of bicycles (used or new) would be exempt just because they are bikes or just because they are used. But then, a VAT of 24% (ours is only 5%) might mean that only a few things are taxed and the tax is not sufficiently broad-based. Anyway, the seller of the used bike will know if he, in his particular circumstance, is supposed to collect VAT from you.

3) The seller proposes not to collect (or remit) VAT, even when he's supposed to, if you pay in cash without a receipt. The higher the VAT, the greater the incentive to cheat, but, perhaps, the greater the penalties for getting caught. Here, because VAT evaders are usually evading income taxes also, the government goes after noncompliant sellers more than buyers -- the seller is still obligated to remit the tax even if s/he never collected it -- but I don't know where the emphasis of the Italian Financial Police lies.

Remember that outdoor piazza-type sales events sometimes contain elaborately choreographed distractions to help pickpockets ply their trade.

If you don't have a receipt for an expensive bike, U.S. Customs might get suspicious and assign a value to it much greater than what you say you paid for it.