Is it anyhow common among native English-speakers to write sentences without subject? I'm not referring to passives but rather to sentences where I am the subject.
My brains are trying to tell me that it is grammatically incorrect but for some reason it doesn't sound wrong.
If you have an example, that would be helpful.
To my knowledge (I never studied grammar further than what the nuns taught me), it would be grammatically incorrect most of the time. Colloquially, I might look over at a friend as I leave the apartment and say "Going to the store." Since I am saying it, and I'm the one in the act of leaving, it would be presumed that I'm the subject, so it would make sense. But if my friend was the one leaving, I could say the exact same words as a question "Going to the store?" and again the subject would be implied. In both cases, however, I don't think it's grammatically correct.
On the other hand, commands can be given without a subject, and they are grammatically correct. So, "Go to the store!" or even "Go!" are perfectly acceptable as complete sentences.