Multiple instances in Linux Mint 18.3 64bit
When wire-desktop seems to spawn more instances every time you make a call. When first starting wire-desktop you have one instance listed in the sound application tab. Each call seems to spawn another instance. Eventually sound will quit working.
Right click the sound speaker on the launch panel and choose 'sound preferences' and then go to the 'applications' tab and you will see multiple instances of wire-desktop listed. As soon as enough instances show up sound will quit working globally on the computer. Restarting wire-desktop will clear out the applications tab and sound starts working again. I have noticed this on several versions of Linux Mint and does not appear to be relative to just one specific release.
Steps to reproduce: Start wire-desktop Make multiple calls over a period of time
After enough calls are made the ring sound when calling someone will fail and then eventually all sound on the computer will quit when enough calls are made.

I am suprised that I have not seen any mention of this in the bug reports yet - at least I have not found them.
System info:
Linux Mint 18.3 Mate Desktop
Linux max 4.14.9-041409-generic #201712251541 SMP Mon Dec 25 15:42:18 UTC 2017 x86_64 x86_64 x86_64 GNU/Linux
PulseAudio

Let me know what other info you need. Really enjoy wire-desktop :)
I believe to have the same behavior on arch linux using wire together with pulse audio. I didn't do more than one two calls per sessions, but even with the first call the pulseaudio helper pavucontrols shows be more than one output device relating to wire.
Actually, only one of them is related to the call, adjusting the volume of the other sliders doesn't affect the volume.
When I close wire all playback devices disappear, ending the call doesn't make them vanish here.
Same on Slackware with multiple wire-desktop/Slackware versions. Also looks like it's related to #598 .
Probably same on Ubuntu 18.04 #1682
Was just about to open a bug report about the incoming call notification not ringing on my laptop, but I think this is exactly my problem. If I pop open Pavucontrol I've got a good selection of seemingly unused Wire instances/streams hanging out.

Besides the call ringing notification, the "call failing" beeps are silenced. The outgoing call sounds and the pinging/incoming message notification sounds are fine for the moment.