I am not a Japanese native, but a Korean. Korean is similar to Japanese, but in Korean, only a form equivalent to "が/は要らない" is used. In Korean, a form like "を要らない" is considered incorrect, because 要る is an intransitive verb, and the subject of the verb is the thing being required.
If you see the Japanese dictionary, all the examples are using が or は.
「燃焼させるには酸素が－・る」 「そんなに時間は－・らない」 「金が－・る」 「もっと人手が－・る」
As a Korean, the "を" form bothers me (even with other verbs and adjectives), but I have seem that form being used in Japanese animations quite a lot, so I do not think it is being considered wrong in Japan. But I would avoid that "を" form if possible.
I would have said "あの人には私が要らない" or "あの人は私を必要としない".