Is there a way to tell when 私が is omitted?

I have a weird question, but here it is. How do we know that in a sentence いつも行くきっさてんのコーヒーはとてもおいしいです the 私が part is omitted?

We had a boy during our Japanese class stumbled upon the translation of this sentence. He knew that café didn't move by itself, but it just didn't click for him that it was a part of the description of coffee.

What if there's a sentence like 私が世話をしている祖父は、面白い話をよく聞かせてくれる。And 私が part is omitted? Or it can't be omitted in sentences like that?