I'm a genealogist with almost 40 years of experience. I did a tree for the Podesta brothers. I found their mother's obit (Mary Kokoris 1918-2007) which only mentions John and Tony Podesta. From that I would assume that they don't have any siblings. Tony was born in 1943 and John in 1949, so neither appear in a public census. The 1950 census won't be public until 2022.
John and Tony's father, John Anthony David Podesta (1910-1980) first appears in the 1910 census with his parents Antonio Podesta (b 1872 Italy - d 1949 Chicago) and Mary Frances Gazzolo (b abt 1879 Genoa, Italy - d 1956 Chicago). The census was taken in April and John was born in January of that year, so he was just a few months old. John appears with his parents and siblings in two more census (1920 and 1930). In 1940 he appears living in his sister Della's household, with her husband Mitchell J Slove. In that census, Della was 38. In both that census and the 1930 census she and her husband aren't enumerated with any children, so it's likely that they didn't have any.
In the 1910 census, Antonio and Civele Mary Frances Podesta have two other children besides John (who was the youngest) and Della. These were Marie and Rose.
In the 1920 census Marie no longer appears. An Illinois index of deaths and stillbirths show that Maria, dau of Tony and Civelle Podesta had died in Mar of 1918. In addition to Della, Rose and John, two more children appear: Lawrence age 7 and Raymond age 2.
The 1930 census show only John, Raymond and Lawrence still living at home. Another daughter, also called Maria was born on 9 Jul 1920, but she died the same day or was stillborn.
In the 1930 census, Rose appears with her husband, Al Lisiandrella and their two children, Katherine age 5 and Michael age 3. I was unable to find this family in the 1940 census, but further research shows the husband to actually be Joseph Lisiandrello (b 1904 Italy - d 1964 Chicago) and the two children to be Katherine Adele Lisiandrello (1926-1976) and Michael Anthony Lisandrello 1926-1999). These two would be first cousins to John and Tony Podesta.
Lawrence Podesta (1912-1982 both in Chicago) married Della M. Mecca or Meccia (b1927 in Chicago) I was unable to locate any more information for these two. Incidentally Lawrence and his brother Raymond both appear in their parents household in the 1940 census.
Raymond Edward Podesta was born in Chicago in 1918. He died in Newark, Licking Co., Ohio on Christmas Eve 1996. I was unable to find any further information about him, regarding a marriage or children. He enlisted in the Army 19 Jun 1941 and was listed as single, without dependents. He finished his military service in 1943.
So, I was unable to find a link between Don Podesta and John and Tony Podesta. That doesn't mean it's not there, just that it's beyond my ability to investigate from home.
If anyone wanted to look further, they could look for obituaries for Raymond or Lawrence Podesta and see if any children were mentioned there. Also, a trip to the Cook County, Illinois courthouse or Licking Co., Ohio courthouse might turn up some additional info.
I would start by looking up their death certificates. Often the informant is a child of the deceased. (Lawrence died 12 Aug 1982).
For Lawrence, who married Della in Cook Co., Illinois on 21 Apr 1954, looking for children named Podesta born starting 6 mos or so and continuing on for 10 to 15 years forward might turn up something, if these records are open to the public. Doubtful because Cook Co. is such a metropolitan area.
There's a marriage for a Raymond S Podesta (not E for Edward, but that may be a transcription error) can be found in the Cook Co., Illinois marriage index. This person married a Mildred Grek on 26 Feb 1944. The marriage license number is D86FBC5F-C173-46F1-A982-C42400080632.
I just found an obit for Della Meccia, wife of Lawrence Podesta and John and Tony Podesta's aunt by marriage. Apparently Della and Lawrence "Larry" Podesta only had 2 children, both girls. They're named as Della Podesta wife of the late Dan Cunningham and Angela Podesta, wife of Dan Sepke. Della and Angela are John and Tony Podesta's first cousins. Here's a link to the obit: http://www.legacy.com/obituaries/chicagotribune/obituary.aspx?pid=181196252
I think I found Raymond E. Podesta's family although the link is a little tenuous. I found a Raymond Podesta who is living in Newark, Ohio. He is related to a Mildred (remember Mildred Grek?) and a Kristi and Michael Podesta. Well Kristi is Kristi Kaufman Podesta, wife to Ray Podesta, and mid 60s insurance agent living in Ohio. His brother, Michael C. Podesta, age 74, lives in VA and is married to Bette, age 79. They have 2 sons: Joshua age 42 and Alfred age 44.
So, unless Don Podesta is related to John Podesta further back than a 1st cousin or 1st cousin 1x removed, there doesn't appear to be a link.
I'm new to VOAT so keep that in mind but if I can I'm happy to show the census or any of the other records as a jpg. I've made my tree hidden and private but can send anyone who wants to see it an invite. Keep in mind that you have to have a subscription to ancestry to see all of the records attached to the tree, but anyone can get a free, two week subscription.
Here's the family again. Antoinio Podesta is John and Tony Podesta's grandfather.
I. Antonio Podesta
B: abt 1872 in Chiesanuova di Ne, Near Genoa, Italy. D: 26 Mar 1949, Cook Co., Illinois. Married 19 Apr 1899, Cook Co., Illinois
Cevile Mary Frances Gazzolo B: 12 Aug 1879 in Genoa, Italy. D: 22 Apr 1956 in Cook. Co., Illinois
A. Maria Podesta (b 1900 d 1918) I think she was most likely unmarried when she died, therefore no issue. The death record I found has her surname as her maiden name, which is why I think she was unmarried.
B. Della Podesta (b 1902 d 1970) m. Mitchell J. Slove I don’t think this couple had children. They married sometime between 1920 (when she was listed with her parents in the 1920 census) and 1930. The appear in the 1930 census as a married couple, and again in 1940, but no children are listed with them in either census.
C. Rose M. Podesta (1906 – 1975) m. Joseph “Al” Lisciandrello
Katherine Adelle Lisciandrello (1925- 1976) m. Harry McCarthy
Michael Anthony Lisciandrello (1926 -1999) m. Virginia Hovey
a. Maria Lisciandrello m. David Kohler
i. Claire Kohler
ii. Michael Kohler
b. Anthony Lisciandrello m. Hugette ¿
i. Virginia Lisciandrello
ii. Natalie Lisciandrello
c. Mickey Lisciandrello m. Dennis Young
d. Peggy Lisciandrello m. Dean Schipke
i. Michael Schipke
D. John Anthony David Podesta (1910 – 1980) m. 24 Jun 1942 Mary Kokoris ( 1918 – 2007) dau of Louis George Kokoris and May L. Kalanze
Anthony “Tony” Podesta b 24 Oct 1943 Cook Co., Illinois
John David Podesta b 8 Jan 1949 Cook Co., Illinois
E. Lawrence Podesta (1912 – 1982) m. Della Meccia
Della Podesta m. Dan Cunningham
Angela Podesta m. Dan Sepke
F. Raymond Edward Podesta (1918 – 1996)
Raymond Podesta m. Kristi Kaufman
Michael C. Podesta m. Bette
G. Maria Podesta born and died 9 Jul 1920 Cook Co., Illinois m. Mildred Grek