I know that in several languages (including Romance languages like Italian, Spanish, and Portuguese) it's not always necessary to include pronouns, because it's clear through the verb form. But it occurred to me recently that I've never seen this in French. Granted, the unique nature of spoken French means that it's generally helpful to have those pronouns, but is the dropped pronoun a thing that ever happens in French?
I don't think so. According to Wikipedia https://en.wikipedia.org/wiki/Pro-drop_language, French is the only exception of the Romance languages.
Grammar terminology for you: a pro-drop language is a language that drops pronouns, for example Slavic languages like Serbian.
But for the French imperative (when you command someone to do something) the pronoun is sometimes dropped. eg Allons-y for Let's go. Despite this, linguists consider French a non-pro-drop language.
Same with English, considered a non-pro-drop language even though the pronoun is dropped in imperatives and other kinds of sentences. eg Come here! for the imperative. In texting, people often drop the pronoun so they say "Going to the shop" instead of "I'm going to the shop".
In spoken French, we sometimes drop pronouns that aren't necessary like :
Il faut aller faire les courses. -> Faut aller faire les courses. (We have to go shopping.)
Il vaudrait mieux calfeutrer la porte. -> Vaudrait mieux calfeutrer la porte. (We'd better caulk the door.)
Il faisait beau hier, hein ? -> Faisait beau hier, hein ? (The weather was nice yesterday, wasn't it ?)
Il y a un type bizarre devant la baraque des voisins. -> Ya un type bizarre devant la baraque des voisins. (There's a strange guy in front of the neighbours' house.)
So yeah it's not correct in written French but in spoken French almost everybody speaks like that.
Are you talking about cases like je sais and not je le sais?