MQTT Transport Binding Item Does Not Push

This is my .items:
Switch doorbell "Switch 1" (all) {mqtt=">[octopus:openhab/doorbell:command:on:default],>octopus:openhab/doorbell:state:*:default],>octopus:openhab/doorbell:command:off:default]"}

This is my .cfg:

Now, if I well understood, the item should publish outbound MQTT packet when state change. So, if a merely did:
curl --header "Content-Type: text/plain" --request PUT --data "ON" http://localhost:8080/rest/items/doorbell/state

the item doorbel shall publish an outbound message, shouldn’t it?
I see the state change in openhab logs

2016-07-25 16:12:22.383 [INFO ] [runtime.busevents             ] - doorbell state updated to ON
2016-07-25 16:12:24.527 [INFO ] [runtime.busevents             ] - doorbell state updated to OFF

but no mqtt message has been receiving at the broker. By using another chanell (e.g. mosquitto), If I subscribe to the openhab/doorbel channel nothing is reveived.
As broker, I used mosca with username/password authentication.

In the examples on the binding wiki page, the “on” and “off” should be uppercase (i.e. ON, and OFF).

I also notice you are missing the “[” on the second and third bindings.

Did you intend the “state:*:default” to be incoming (i.e. messages published on openhab/doorbell changes the state of the doorbell Switch)? If so the direction is wrong, it should be “<”.

Finally, you might need to use “autoupdate=false” to prevent the incoming messages from causing a new command being published back to the topic resulting in a loop.

Switch doorbell "Switch 1" (all) {mqtt=">[octopus:openhab/doorbell:command:ON:default],<[octopus:openhab/doorbell:state:*:default],>[octopus:openhab/doorbell:command:OFF:default]", autoupdate="false"}

There likely were some errors printed to the logs complaining about not being able to parse the binding configuration string with the errors described above.

Thank you for your reply. Yes, there was an error in my copy&pasted code. However, I checked and it was fine.
I tried with your suggestions:

Switch doorbell "Switch 1" (all) {mqtt=">[octopus:openhab/doorbell:command:ON:default],<[octopus:openhab/doorbell:state:*:default],>[octopus:openhab/doorbell:command:OFF:default]", autoupdate="false"}

The logs in the shell:
07-25 20:08:01.448 [DEBUG] [m.i.MqttGenericBindingProvider] - Removing message publishers for item doorbell
2016-07-25 20:08:01.449 [DEBUG] [o.i.t.m.i.MqttBrokerConnection] - Unsubscribing message consumer for topic ‘openhab/temp’ from broker 'octopus’
2016-07-25 20:08:01.520 [DEBUG] [b.mqtt.internal.MqttItemConfig] - Loaded MQTT config for item ‘doorbell’ : 1 subscribers, 2 publishers
2016-07-25 20:08:01.520 [DEBUG] [o.i.t.m.i.MqttBrokerConnection] - Starting message consumer for broker ‘octopus’ on topic ‘openhab/doorbell’

If I do:
curl --header "Content-Type: text/plain" --request PUT --data "ON" http://localhost:8080/rest/items/doorbell/state

(Does that command simulate a state update ?)

Basically, I would like to publish a message every time the device state changes (I am using a websocket to subscribe to the item’s channel/topic).

You will have to do that in a rule. A message will only be published to MQTT when the Item receives a command. That is actually the key distinction between an update and a command, updates do not trigger the binding.