Unless it's one of the "disassociated by actions" special cases then there has to be a letter of disassociation from the individual to the congregation. Of course an acknowledgement would be polite but the individual has taken the formal step of documenting their wishes so I don't see that the lack of any further correspondence is a problem.
The other thing is that there will be a public announcement at the next meeting.
In some cases the elders may try and speak to an individual if they think there is some chance their mind can be changed or if perhaps there are some other circumstances specific to the case but in my experience most people that DA themselves have absolutely no intent of being any part of the org and will already be considered bad pennies. Most of the time the BoE will be only to glad to be shot of the person.