Siena, like Florence, will have lots of music concerts across the city.
The main one will be in piazza del Campo starting at 9.15pm, with
Deschema, a local band, then
Alex Britti, an Italian pop star until midnight, free entry.
There will be swing music in
Piazza del Mercato with from 9.30pm, free entry.
Piazza Indipendenza will have an ensemble of a bassist and 8 percussionists paying samba + electronic beats from 10.30pm, free entry.
In
Piazza Tolomei there will be Caribbean sounds, salsa, merengue, mambo and bachata with a Tuscan band called Mambo Kids, starting at 10.30pm, free entry.
I found a private tango party at the
Teatro dei Rinnovati with Oblivion Tango starting at 10.30pm, paid entrance so here you need to book ahead.
I got this info from La Nazione article on the events going on in Siena:
https://www.lanazione.it/siena/cosa fare/capodanno-2019-1.4360986
If you're looking for a dinner + party at a restaurant, you really have to go by the restaurant and finding out what they are offering - usually a set menu with set price that starts at dinner and goes on until midnight.
Happy start to the new year!