Android
Turn On/Off Outgoing Message Sounds auf Android
The click you hear after sending a text comes from Google Messages itself, not from your phone's ringtone or notification settings. That is why muting your device or switching on Do Not Disturb has no effect on it: the sound is generated by the app's own confirmation tone, separate from your system audio profile.
This confirmation sound and the sound for incoming messages are controlled by the same switch inside the app. Turning it off silences both directions at once. There is no separate option to mute only outgoing messages while keeping the incoming sound active.
To change it: Tap on Messages ➔ Tap on Google Account ➔ Tap on Messages settings ➔ Tap on General ➔ enable or disable Hear outgoing and incoming message sounds.
If your goal is to silence notification alerts for specific conversations rather than the confirmation click, this toggle is not the right tool. Use per-conversation notification settings or a Focus mode instead, since those control alert sounds and vibration, while this switch only affects the short confirmation tone.