However, I don't think you would write pisze on its own, simply, once it doesn't give away who you are talking about (the subject). So, assuming you would need to mention the subject in case you talked about the 3rd person, I believe it would be known while writing even if you didn't write the letter correctly.
In other words, I mean, if you are talking about the 1st person, you'd either write it like "piszę" and incorrectly like "pisze", and people would still assume you're talking about the 1st person because you didnt make the subject clear!
Now, even incorrectly writing, if you were talking about the 3rd person, I tend to think you would specify the subject, and therefore, it would be still understandable.
On/Ona pisze or piszę.
But don't write it wrongly. It's better to avoid these situations by even specifying the subject, in case you're not sure of how to write it.
Every single day, I read a few comments claiming that a correct (even starred) answer was rejected. It seems impossible that all of those people made bigger typos, there clearly must be some bugs. Although I don't think I personally had any correct answer rejected in any course...