The two certainly originated independently. Blackletter started elongating the medial s in the 8th century (Long s); Greek started using the pre-8th century lunate sigma as a final form, from the 11th century on (Letters). Both Greek and Latin scripts invented lowercase at the same time, but there was no real cultural contact between West and East until the 1400s, so this seems to be a happy coincidence.
Wikipedia speculates that the existence of a positional variant in Greek sigma reinforced the persistence of a positional variant in Latin s. I’m not convinced; everyone was using the long s, and had been for a while before the invention of printing.
EDIT (from comments):
When Lowercase was invented in Greek in the 8th century, Σ was long dead (it’s an 18th century revival). The uppercase was Ϲ, and the lowercase became σ, which was used everywhere: Ὀδυσσεύσ.
Someone in the 11th century had the idea that the final sigma would look nicer as the original lunate, rather than curling inwards: Ὀδυσσεύϲ. I can kinda see that: it’s a symmetrical form at the end of the letter; it lends itself more readily to ligatures (e.g. οϲ => something that looks like Co attached at the top); the pen leaves the page at the bottom of the line, not mid-line (so less smudging). A few centuries later, the ϲ got a tail added to it, and became ς.
Something quite different happened in Frenkistan (I love that word), and I don’t get it. I don’t get why all of a sudden an s in the middle of a word in the 8th century would turn into a ſ. I can see the ductus argument for Greek, but not for German. Whatever it is, is wasn’t the same thing.
But Greeks in the 8th century and Germans in the 8th century were using the same parchment and the same pens; so while the details of the scripts might have been different, the pressures on what letters would look like must have been similar.