As other posters described: Lead is not soluble in water with neutral pH (near pH of 7). However, rain water exposed to carbon dioxide in the atmosphere has an acid nature - pH=5.6. The higher the carbon dioxide, the more acidic the atmospheric water. Lead becomes much more soluble at acidic pH.
That said, lead bullets sitting in the shallow soil is not a significant threat unless the water table is just below the surface.
Lead shot sitting in very shallow water in swamps and wetlands is not a threat unless wetland birds pick up the shot while grazing on the bottom. Eagles and other birds of prey hunt the wetland birds and ingest the lead shot picked up in the prey birds. Stomach juices are VERY acidic (pH=1) and can dissolve the lead pellets, sending that lead into the blood stream. As Marty said, several eagle species were near extinction due to lead poisoning (and some other environmental toxins). With controls on lead shot, the eagles have recovered to reasonable population levels.
Lead did impact the baby boomer generation and the effects of lead toxicity has been documented all over the world. During the period of the use of leaded gasoline (1920's to late 1960's), children in communities exposed to high levels of motor vehicle traffic showed concentrations of lead in their blood that cause brain damage, learning disabilities and behavioral problems. Russia built industries that discharged high levels of lead into the air in former countries of the old Soviet Union. Children in cities of Poland with those industries showed toxic levels of lead in their blood as well as elevated incidence of learning disabilities. Up to 80% of children in those lead contaminated areas had asthma.
There is 60 years or more of research on lead toxicity and 1000's of papers in that literature. In 1897, Australian researchers identified lead in paint as the cause of a "Toxicity of Habitation," and the first US. case was reported in 1914. By 1917, U.S. medical authorities had established that childhood lead poisoning from lead paint was a common problem. "A child," wrote a medical commentator in 1924, lives in a lead world." Toxicity of lead was brought up in 1862 in Germany.
Ron