Psyonix pushed a server-side update today that enables PlayStation players to matchmake with players on Xbox and Nintendo Switch, in addition to the always available PC crowd. You do not need to download anything for this to work, just join a match like you would normally.
The one thing you still cannot do is create a cross-platform party and enter matchmaking. That is coming very soon, however. According to Psyonix, the first update of 2019 will include the Rocket League ID system that allows you to form a team and go into public matchmaking. That said, you’ll need your own voice solution. In a comment on the PlayStation Blog, Jeremy Dunham from Psyonix said voice chat in cross-platform parties is not in the planning, and said it’s by design due to their “E for Everyone” rating.