Both teams used Tuesday's event to continue to build their rosters ahead of entering MLS as expansion teams in 2020.
Sweat, a 28-year-old left-back, joins from New York City FC and will return to play in his native Florida. He earned his first two U.S. national team caps in 2018.
Danladi was the first overall pick in the 2017 MLS SuperDraft but struggled to build on an eight-goal rookie season in 2017, scoring just three goals combined over his next two seasons.
Nguyen will be the headline-grabber in that group, as the 33-year-old has developed a reputation as one of the top creative midfielders in MLS during an eight-year stint in the league.
Those players wouldn't all stay with Nashville for long. The club quickly traded Valentin to the Houston Dynamo for goalkeeper Joe Willis, while they also traded Vazquez to FC Cincinnati for allocation money.
Inter Miami, co-owned by David Beckham, will begin next season at the site of the former Lockhart Stadium in Fort Lauderdale, Florida, while Nashville SC will play at Nissan Stadium, home of the NFL's Tennessee Titans, while their new stadium is under construction.
For Inter, Tuesday's haul of players joined nine players already signed for the 2020 season: Matias Pellegrini, Julian Carranza, Victor Ulloa, Christian Makoun, David Norman Jr., George Acosta, Jay Chapman, Grant Lillard and Jerome Kiesewetter.
Meanwhile, Nashville already had the following players under contract before next season: Hany Mukhtar, Dax McCarty, Anibal Godoy, Randall Leal, David Accam, Dave Romney, Derrick Jones, Daniel Rios and Cameron Lancaster.