When female ticks get engorged they fall off and find a place to hide and lay their eggs to make thousands of babies which is why getting them BEFORE that happens is important.
Sometimes they do leave an irration. More often, not.
I LOVE the cheap tick twister you can buy at Petco. Each time it has fully removed the tick including the mouthparts with no breakage and works in thick hair.
This is a good site.
University of Rhode Island TickEncounter Resource Center