Live virus vaccines have been known to "shed" their viruses.
This means the vaccinated host can be asymptomatically infectious, leading to outbreaks. Some vaccines, like the DTaP vaccine, have still been found to cause asymptomatic illness in the vaccinated host, even if not a live virus vaccine.
Vaccine Package Inserts:
Rotavirus- https://www.merck.com/product/usa/pi_circulars/r/rotateq/rotateq_pi.pdf, https://www.gsksource.com/pharma/content/dam/GlaxoSmithKline/US/en/Prescribing_Informatio n/Rotarix/pdf/ROTARIX-PI-PIL.PDF
Varicella Zoster- https://www.merck.com/product/usa/pi_circulars/v/varivax/varivax_pi.pdf
Yellow Fever- https://www.fda.gov/media/76015/download
"..the detection of VZV DNA in saliva of Zostavax recipients for up to 28 days suggests that contact with saliva of recently immunized individuals represents a potential source of transmission." https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3096786/
"Some individuals may experience shingles or chickenpox-like rashes within 42 days after receiving zoster vaccine. Transmission of VZV virus from vaccinated individuals to other individuals occurs rarely." https://www.medicinenet.com/zoster_vaccine_live__shingles_vaccine/article.htm#which_drugs_or_supplements_interact_with_zoster_vaccine_live__shingles_vaccine
"Prevalence of shedding was inversely related to age, and maximal shedding occurred within 2 days of vaccination." "Shedding had stopped in most cases by 11 days post vaccination." https://www.cdc.gov/flu/professionals/acip/safety-vaccines.htm#SafetyLAIV
In our dose-response study, viremias were found until 36 days after vaccination. Recent studies report frequent (44%) or prolonged (until 198 d) shedding of vaccine RNA in urine of vaccinated individuals. https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4635904/