I believe the concern about swimming with a new tattoo is not about spreading diseases or germs to other people as it is about what it could do to your new tattoo.
From what I understand, It is generally advisable to avoid baths, overly long showers or swimming to prevent the tattoo from becoming soaked and to protect it from bacteria and what-not from infecting the new tattoo since it is essentially an open wound. Strong chemicals and chlorine in the water can also irritate the new tattoo and interfere with the healing process.
Other than avoiding the pool until your new tattoo is completely healed, you may want to monitor it for any signs of infection. Elsewise, just wait for it to heal.