Firstly, chopmarks are not countermarks or counterstamps as such: These were usually marks applied after striking the coin, by the mint or govt agency to mark the coin for some purpose.
Regarding the chopmarks themselves, metal was NOT removed from the chopmark for the purpose of assaying, the chopmark was applied AFTER passing the assayer's tests.
Usually, merchants(mostly money changers or local banks in China) would use a specially trained assayer called a "shroff", to test/verify the silver coin. The most accurate way was to test "specific weight", but this was too time consuming for hundreds of coins on the spot. The shroff used a combination of sensory skills to evaluate the coin: weight, balance, tapping sound, which could take just a second or two, and the coin was then stamped with the chopmark(probably passed to another person to apply the chop).
Among merchants in a particular city/region, their chop marks became known, so if that chop mark was applied, it would be regarded as genuine. If a coin left that city/region, it would reevaluated in a new city/region, and a new chop mark applied.
Chopmarks come in different sizes, but it seems that the size was generally determined by the merchants and what was in vogue at the time: earlier ones were smaller(maybe late 1700s up to mid 1800s), over a period they grew larger(mid 1800s to 1900), then grew smaller (after 1900).
Here is a Japanese silver 1 Yen, with both a countermark/counterstamp, and a chopmark.
The countermark is the circular mark, which designates that the coin was to be exported from Japan for trade (not to be used in Japan for currency). This was called a "gin" or "silver" mark in Japan.
Marked as it is on the left side, it shows that the countermark was applied in the Osaka Mint. If applied on the right side, it would be the Tokyo Mint. (I have other 1 yen coins with the Tokyo "gin" countermark)
The chopmark is to the left of the countermark. (see detail in the second photo)
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