Does sleep-sharing increase or reduce the risk of sids?

Yes: If by sleep-sharing you mean sharing the same bed (often called "co-sleeping"), then the risk of sids is probably increased. Co-sleeping with the baby next to the bed in a safe, side-car type of thing (sometimes confusingly called a "cosleeper") does not increase or decrease the risk of sids.

Yes: A young baby sleeping face-up in a crib without fluffy blankets nor pillows around, without dangerous cords or strings nearby, will be safe there. His chances getting sids are low (998.5-999 out of 1000 babies won't have sids). Sleeping in a bed with an adult adds the risks of suffocation from blankets & pillows, from trapping the head at the mattress' edge, and from being squashed by the adult.

Conflicting research: The research on this issue is conflicting and not clear cut. We do know that breastfeeding mom's reduce baby's risk of sids; we prefer that mom's co-sleep with baby lying in a co sleeper next to parents. Jim Kemp a pulmonologist at Wash University is one of the foremost researchers on the subject. http://www.cribsforkids.org/2012/07/30/bed-sharing-with-infants-is-linked-to-their-deaths/