And a few hours before the NBA trade deadline, they found a way to meet each other's needs.
"Ramon is a proven veteran who will bring an up-tempo style to our second unit and give us quickness, energy and defensive presence," Wizards president Ernie Grunfeld said. "He can make plays for himself as well as others and will help to solidify our bench as we head into the final 28 games of the season and the playoffs."
The 28-year-old Sessions has struggled in Sacramento, averaging 5.4 points, 2.7 assists and 17.8 minutes for a team that is on its third coach this season. But he averaged 12.3 points, 4.1 assists and 26.7 minutes with Charlotte and Milwaukee last season and has been a reliable reserve for most of his career.
The players dealt were among 39 to change hands Thursday, a day that marked the busiest NBA trade deadline day in 25 years.
Washington (33-21) is counting on Sessions to be a more athletic defender than Miller. Sessions also is a shoot-first point guard who can give the second unit some scoring punch.
Miller will be reunited with new Kings coach George Karl in California's capital city. The two had a close relationship in Denver, where Miller enjoyed some of his best seasons.
Karl and the Kings (18-34) will lean on Miller to provide veteran leadership on a team that has sorely lacked it. The Kings hired Karl over the All-Star break to replace Tyrone Corbin, who took over after Michael Malone was fired 24 games into the season.
Miller, who turns 39 in March, is averaging 3.6 points, 2.8 assists and 12.4 minutes. He played for Karl from 2003-07 and 2011-13 with the Nuggets.
"Adding Andre bolsters our backcourt with one of the most prolific ball distributors in the game," said Kings general manager Pete D'Alessandro, who worked for the Nuggets during Miller's time in Denver. "He'll also provide veteran leadership to the locker room and a wealth of experience at the point guard position. We're excited to have him join the team. We also want to thank Ramon and wish him the best moving forward."
Both teams return from the All-Star break Friday night. Washington hosts Cleveland, and Sacramento is home against Boston for Karl's debut as the franchise's coach.
The Kings are also planning to call up guard David Stockton from the NBA D-League's Reno Bighorns for a 10-day deal, according to league sources.
Sources told ESPN.com that the Kings want to bring Stockton in from their D-League affiliate in Reno and give the son of NBA legend John Stockton an opportunity to help them fill a need in the backcourt until the newly acquired Miller settles in.
Stockton, who played at Gonzaga before matriculating to the pros this season but has not yet played in an NBA regular-season game, was briefly a teammate of Miller's in October when he went to training camp with the Wizards.
Information from ESPN's Marc Stein, ESPN Stats & Information and The Associated Press contributed to this report.