An interesting idea, Richie. In most of the Scots texts listed in Child (243 C - F) it's very clear that the opening lines are spoken by the woman, but these usually begin: "Where have you been (my long lost lover etc)", rather than the "Well met, well met" formula common in the US versions.

However, the Distressed Ship's Carpenter version attributed by Child to the 'Rambler's Garland' of 1785, but appearing earlier in 'Diverting Songs' (1730s) - and which is thus is temporally intermediate between the 1657 Laurence Price broadside and those oral versions - does begin with "Well met...", but in this case it's equally clearly the returning mariner who is talking.

The US broadside copy often attributed to 'De Marsan', which is believed by some to be the template for many US versions, follows the same pattern as the (English) Distressed Ship's Carpenter - it's the sailor who says:

Well met, well met, my own true love
Long time have I been seeking thee
I'm lately come from the Salt Seas
And all for the sake, love, of thee

We had a really good discussion of the New England variants of the ballad here, in which Jon Minear supplied a number of texts. Amongst those you could find either protagonist uttering that first line:

Compare, for instance:

"Well met, well met, my own true love,
Well met, well met," said he...

[Belle Richards, Colebrook, NH]

...with:

"Well met, well met, my pretty fair maid,"
"No so very well met," said she...

[Elmer George, East Calais, VT]

The history of this ballad is quite tangled. I've formed the view that the US versions contain DNA from both English printed, and Scots oral versions.