I suspect that the discrepancy flows from the definition of welfare used. There is probably also a life cycle effect, with some kinds of benefits used only at the end of life and terminating in death, and others used pretty much only when one has young children.
The public radio claim is as follows:
Researchers found that over a four-year period, 79 percent of people
who received cash General Assistance - known as "welfare" - left the
program within two years. About half left supplemental security
income, medicaid, and food stamps in the same amount of time.
The major outlier was housing assistance - including public housing
and subsidized rental programs. There, only 40 percent dropped out
within two years.
The CNS claim is as follows:
[T]he 109,631,000 living in households taking federal welfare benefits
as of the end of 2012, according to the Census Bureau, equaled 35.4
percent of all 309,467,000 people living in the United States at that
When those receiving benefits from non-means-tested federal programs —
such as Social Security, Medicare, unemployment and veterans benefits
— were added to those taking welfare benefits, it turned out that
153,323,000 people were getting federal benefits of some type at the
end of 2012. . . .
82,679,000 of the welfare-takers lived in households where people were
on Medicaid, said the Census Bureau. 51,471,000 were in households on
food stamps. 22,526,000 were in the Women, Infants and Children
program. 20,355,000 were in household on Supplemental Security Income.
13,267,000 lived in public housing or got housing subsidies. 5,442,000
got Temporary Assistance to Needy Families. 4,517,000 received other
forms of federal cash assistance.
TANF aid, which is the narrowest commonly used definition of welfare, typically is relatively short term and episodic, as is WIC which like TANF has a built in time limit. These programs have the highest turnover in the study cited by the public radio station.
SSI and public housing tend to be much more long term. The public radio story confirms that the housing program is long term, but shows an intermediate amount of turnover in other programs.
Normally, SSI is paid to seniors or disable people who also qualify for Social Security benefits, can't work, and don't have savings, so participation in those programs tends to be long term.
The waiting lists for public housing assistance are typically many years long, so only the long term poor tend to end up getting it.
Eligibility for food stamps (officially known as SNAP) is materially more generous than eligibility for the cash welfare TANF program and so people tend to be recipients of it for a longer period of time.
Medicaid really consists of multiple programs with different eligibility requirements and different benefits.
For example, one very significant component of the Medicaid program which is rarely characterized as "welfare" is its nursing home assistance program which is utilized mostly by the working and middle classes and provides a majority of the funds for nursing home care in the U.S. This program actually is usually for a short period of time because the average nursing home stay is about one year (predominantly ending in death rather than returning home). Obviously, people who leave the Medicaid program in that manner don't return to the general population afterwards.
Another significant component of the Medicaid program involves people who are permanently disabled and usually receiving some form of disability benefits as well. This is also long term in most cases.
A third component of the Medicaid program involves working age, not disabled people and their children who either receive pre-expansion benefits because they are in or very nearly in poverty, or who receive post-expansion benefits because they are working class. (There are actually some additional subsets of this population, CHIP for children, a pregnant women's program and a program for adults, in addition to expansion and buy-in components). Often people are made eligible for Medicaid on the basis of the same application that qualifies them for food stamps and TANF.
But, lots of people who are formally in the Medicaid program don't actually receive any Medicaid benefits. Some are just healthy people who don't need medical care. Others are people who need medical care but can't find a provider who accepts it (because Medicaid reimburses providers at far below the market rate or the rate paid by Medicare, many providers refuse to accept new Medicaid patients). Definitionally, it isn't obvious that someone eligible for the Medicaid program who uses it when they get step throat or a urinary tract infection once every three years, for example, is a continuous or an episodic beneficiary of this means-tested program.