My skin got itchy after starting T too. I have eczema, a non-contagious skin condition that causes rashes to break out in itchy patches on my body. I was worried that the increased body hair growth from T would make my eczema worse. The hair growth definitely made me itchy (The areas with new hair growth seem to be the itchiest areas). Occasionally, the itching and irritation would make my eczema break out, but I have medication to keep that under control.
I've been on T for almost four years now. My skin is still itchy, but only in the areas with new hair growth. The places on my body where hair has been growing at a regular rate for a few years no longer itch.