Why do Emiratis sometimes drop pronouns completely?

This is not specific to Emirati Arabic. Verbs in Arabic has a unique conjugation for each pronoun, well mostly. Because of that, the pronoun is understood making it redundant.

In the example you gave for أسير, is only conjugated for the pronoun أنا (I)

P.S. The correct translation to أسير السوق is: I go to market. To say I am going to the market, you would say:

• أنا ساير السوق / ana sāyir essūq /

We use something called essm al-fāʕl. This is a whole topic by its own.