Depends on what type of poison first of all,and whether its external or internal.Poison ivy for example is more of a direct contact reaction irritation rather than a contagious pathogen epidemic.Salmonella food poisoning can be spread as live bacterium pathogens through direct contact with infected persons body fluids.
Yes, if the poison is not washed properly from infected person's clothes, shoes or body then it can be transmitted to other person. But still it depends upon what type of infection it is.