Messier and Mechain had essentially the same technology at their disposal, astrolabes (the real workhorses of astronomers of that period) and modest aperture long focal ratio telescopes. Realistically useful large aperture and short focal ratio (better for light amplification) telescopes did not become practical until the late 19th century. (There were a few earlier examples, but their use was very awkward.) Mechain had basically the same goal as Messier, discovering comets, the same technology, and the same problem of avoiding confusion with existing permanent objects that kind of looked like comets.
The methodology of the day mostly depended on naked eye sky scanning, using an astrolabe to accurately measure the position of a target object in the sky. The small telescope might be used once the object was spotted to see if it could be resolved into something more interesting than a diffuse smudge. (For the most part they couldn't, but the telescope could make a cometary tail visible much earlier than it would be visible to the naked eye. This sort of observation was normally only attempted after the object was located and identified.)
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