Here in the US, I’d always thought we were borrowing a British expression, but when I was trying to check on the web just now, it looks more like a phrase that we are fond of and others don’t make much use of.
I’m 62 and have lived in England all my life.
I’ve rarely heard ‘both sides of the pond’.
A little more frequent is ‘the other side of the pond’ - because there are some interesting differences between the UK and the US.
For example “He dropped some chips on his pants” means:
(UK-US translation) he dropped some fries on his boxers
(US-UK translation) he dropped some crisps on his trousers
I think you lot over the pond got it from us over here.
Whilst referring to the Atlantic as “the pond” is universally understood, it is now more than a little quaint and someone using it in the wild would be quite odd.
I am guilty of using it - even on this erudite forum, but I do acknowledge that it, like me, is a little old fashioned.
Well all right, then may I still refer to GB as “Old Blighty” ? 
The ‘Old’ is rather redundant these days, but ‘Blighty’ is fine.
“Pond” as a (usually humorous) alternative to “ocean” originated in Britain, and could refer to oceans in general, or any particular ocean. It only referred to the Atlantic Ocean specifically when the context required that. But over time it came increasingly to refer to the Atlantic, and became increasingly a characteristic US usage. In BrE it would be considered so old-fashioned as to be mannered, I think.
And, yes, it’s rarely “both sides of the Pond”; more usually “the other side of the Pond”, since the expression is mostly used to point to differences between British and US language, habits, culture, etc, not to similarities.
I think “across the pond” is the most common way it’s phrased and it has its own Wiktionary entry as such.