No. After exposure to air, STDs die pretty quickly.
In theory, you'd have to use a sex toy on someone and then immediately use it on someone else to transmit an STD. In practice, it's not a terribly efficient way of transmitting STDs. Can it happen? Sure. It just doesn't really unless it's a pretty immediate share.
However, other bacteria and fungi and such can grow on a toy depending on the material of it (how porous it is), how it was stored, etc. It's good to clean your toys after each use, even if it's just you using them. Follow the instructions from the manufacturer.