Not quite. The first part actually doesn't make sense. In your 2nd sentence you are using the -ing form to make a reduced relative clause.
Here's the problem: relative to what? Relative clauses usually refer to a word (or a phrase) in the main sentence, and they use the relative pronoun (such as who) to make the connection. A reduced relative clause omits the relative pronoun, but keeps the (intended) meaning.
In your example, the main clause is:
We were seeing rain dropping*
The problem is, that the reduced relative clause is often found next to the word it refers to. Like in your other clause (which is the part that you handled correctly):
standing next to each other at a bar without saying anything
this clearly refers to "we"
We were standing next to each other. (At the same time) we were seeing rain dropping. -> Standing next to each other, we were seeing...
The clause in question stands before that one, and since there isn't another finite clause in between it also refers to "we" - and this is where the problem is. The fact that the rain is falling doesn't refer to "we" or any other part of the main clause.
What's more - you kept the subject (rain) of the sentence you were trying to reduce - and the whole point of these clauses is omitting parts to avoid repetition.
To convey what you wanted to say:
While the rain was falling...
but then you repeat that you actually watched the rain, which makes the whole sentence a bit off (stylistically at least).
There are endless ways to fix it. One might be:
Watching the rain outside and running out of things to talk about, we stood next to each other at the bar, without saying anything/without a word/.
all the same except:
we stood silently next to each other at the bar.
(and silently can go to several places, you can play with it for emphasis).
*This one, btw isn't idiomatic. You could see the rain dropping (falling, dripping?) but if you were to do something you would watch it (action). If you want to say that you were able to see it you would use a modal verb (could e.g). If you want to say that you simply have observed the fact that the rain is falling, you would use a different tense (not a progressive one for sure).