I have heard people replacing 私、俺 with こっち in casual speech. For example: こっちは忙しいんだ instead of 私は忙しいんだ. The question is, under what circumstances can one use こっち instead of 私. I know that そっち can mean "my side", which might imply that there may be more than 1 person on my end
Similarly, in using そっち instead of あなた (or the person's name)、are there any differences besides そっち meaning "you and anyone on your side" whilst あなた meaning just you(and no one else)?
こっち or こちら as a substitute of I/we can have a tinge of rudeness to it as it is used to emphasize physical or psychological distance.
The form こっちはXXX (I am XXX) is usually used to expresse frustration or grievances (こっちは忙しいんだよ！こっちは寝てないんだよ！こっちは大雨なんだよ！ etc). E.g.:
明日取りに来いだって？こっちは大雨なんだよ。ふざけんな！
こちら is also used frequently to imply "we don't care about your situation", E.g.:
そちらも事故への対応で大変だとは思いますが、こちらとしては契約を守っていただかないと。
There are neutral use cases like the following:
こちらは平気ですので、ご自身の勉強に集中なさってください
こっちも大変だが、向こうも大変だ
But even then it implicitly creates "sides". So, unless you want to create this effect, I wouldn't do it.
Credit: Answer was updated based on @naruto's comment.
