It's simply an abbreviated subordinate clause. We leave out "to be" and the subject, which are the same tense and subject as in the main clause. In Spanish,, we can do something similar, but without "while" (which is understood).

Caminando por la calle, hablo por teléfono.

A more common construction in Spanish would use the infinitive, nominalized by the definite article:

Al caminar por la calle, hablo por teléfono.

In any language, it would be clearer to state the subject in the first clause:

I talk on the phone while walking down the street.

Thanks for the feedback, Hive Mind! This is my first time posting so I'm not sure how to comment on your individual responses. So...

Marianne: Yes, she understood the meaning of the phrase by the context, but not the reason for the sentence construction, i.e. why words are missing. We talked about the phrases being understood to mean "while [I was] walking." Thanks for your input.

Lance: I will pass on your remarks to my partner. She is a journalist and very interested in the details of the language.

Phil: Very clear explanation. I will also pass this on to my friend. I think showing the Spanish comparison will help her understand.

I'm curious - do they get the missing "I was" in your version? Since you can use the words "I was" in both versions, would it help them if you pointed that out? Just a thought.

Sometimes I think English was designed to make people crazy. ;-)