I'll mostly answer this question at the hand of lead and to a lesser extent silver recycling.

Simply put metal recycling was very common in not only the Roman era but long before and after the rise and fall of the empire. To refine your question we will be looking at the provenance of input material for the metal forges where this recycling would have taken place. To do this we usually look at both the ore, metal and slag remains at the forges and the end products.

In the case of lead and silver this would have been done with lead (Pb) isotope analysis. The reason this also relates to silver is that the main source of silver in the Roman empire was Galena which is composed of both Lead and Silver. When objects are created from a single source of ore we are able to locate the geological origin point of the metal through its unique isotope signature. While this would be an outcome we would like this is not the one we most often get. When metals from multiple origins are mixed the Pb isotope ratios go out of whack and it is quite quickly clear it was recycled. We can find this when analysing both forge remains and metal objects themselves.

Another way we know of lead and sometimes silver recycling is the presence of tin (Sn) within either the forge or object. Because of the rather cheap nature of tin (after the bronze age) it was used as an additive to silver and lead. This might have been to dilute it or to make it more malleable. In either case this indicates clear recycling and it is found rather extensively throughout the empire.

While extensive literature about the phenomenon of metal recycling in the Roman Empire is rather scarce, every excavation report of a forge in the past half a century have included isotope analyses for ore provenance and with most of them there is a side note of the presence of recycling. As this is quite often a dead end within the trail of archaeological clues, it isn't followed up upon.