Not in that way. It would be:
"I caught sight of him in the street"
"He caught sight of me in the street"?
One can say, 'I caught his eye.', meaning, I attracted his attention. This, then, could be meant in two ways.
The slinky blonde sashayed into the room and caught the eye of every man there, single and married. (Everybody noticed her even though she didn't say a word, and wasn't trying to catch anyone's attention, catch anyone's eye in particular.) Funny that.
I waved my hand to him across the room until I finally caught his eye. (He did not notice at first, then saw me waving - I caught his eye.)
Which all brings up the question, when do we use 'in the street' and when do we use 'on the street'?