Could Gollum see Frodo? I don't think he could.
He could smell Frodo, certainly (no bath in how long???) and no doubt he saw exactly where Frodo was when he put the ring on.
And I think (guessing here, but bear with me) that Frodo would have been far too focused on finally putting the Ring on and wearing it, and experiencing all that that entailed, to move far from the spot. He wasn't putting the Ring on to hide, so why would he have moved away from the spot that he was standing on?
For me, it adds to the suspense that Gollum was "wrestling an unseen foe", because I want Gollum to "win" and get that dratted thing off of Frodo.
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...down to the water to see the elves dance and sing upon the midsummer's eve.
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