What is Right Action for a Buddhist regarding talking about personal success? In this question 
a comment stated in part "The old sense of transferring merit had to do with mudita, you tell other people about your success so that other people can be happy about that. Being happy about other people success is meritable, so it works like transfer."

The idea that telling people about your success is wholesome, is very surprising to me. I somehow had the impression that from a Buddhist point of view, being very modest and not making such things known was better.

Did the Buddha teach anything specific about this? Thank you.
I would suggest that it depends on how you relate to the success, your attitude to it. Relating an experience can be beneficial to the recipient in many cases, both in terms of use as a teaching and possibly of merit as you mention. In terms of telling about a successful experience as a teaching, this can be related to the Udayi Sutta in the Anguttara Nikaya, on how to teach the Dhamma. 


  "It's not easy to teach the Dhamma to others, Ananda. The Dhamma should be taught to others only when five qualities are established within the person teaching. Which five?
  
  "The Dhamma should be taught with the thought, 'I will speak step-by-step.'
  
  "The Dhamma should be taught with the thought, 'I will speak explaining the sequence [of cause &amp; effect].'
  
  "The Dhamma should be taught with the thought, 'I will speak out of compassion.'
  
  "The Dhamma should be taught with the thought, 'I will speak not for the purpose of material reward.'
  
  "The Dhamma should be taught with the thought, 'I will speak without hurting myself or others.'
  
  "It's not easy to teach the Dhamma to others, Ananda. The Dhamma should be taught to others only when these five qualities are established within the person teaching."


Interpreting personal success as in the interest of self promotion is not helpful in terms of the teaching - so again it comes back to the attitude towards what is told. If for the benefit of the recipient (in terms of the Path), it could certainly be helpful. 

Mudita as a divine abiding is a wholesome state, and allowing mudita to arise in oneself and others by sharing a success experience would be beneficial. However, modesty and humility are beneficial traits to acquire, especially for those of us who have not entered the stream. For most of us, sharing success stories is more about self inflation than for the benefit of others, even though sometimes on a subtle level. 

So this answer is 'it depends' - and hopefully the five guidelines above can be of help, when deciding whether to convey a success story, and how to.

(edit: A few notes on Right Action and Right Speech, since they're mentioned in the question)

Right Action

The question refers to Right Action - samma kammanto in Pali - which is part of the Noble Eightfold Path. Specifically, Right Action refers to action that is in accordance with three of the five precepts - namely abstaining from the taking of a life (i.e. not killing any living being), abstaining from taking what is not given (i.e. not stealing), and abstaining from sensual (or sexual) misconduct. I do not interpret sharing a success as related to Right Action. However, in communication, Right Speech may be more relevant.

Right Speech

As translated on ATI:


  "And what is right speech? Abstaining from lying, from divisive speech, from abusive speech, &amp; from idle chatter: This is called right speech." 


This may be helpful in deciding what to say, when to say it, and whether to say something at all:


  "Monks, a statement endowed with five factors is well-spoken, not ill-spoken. It is blameless &amp; unfaulted by knowledgeable people. Which
  five?
  
  "It is spoken at the right time. It is spoken in truth. It is spoken
  affectionately. It is spoken beneficially. It is spoken with a mind of
  good-will."


With metta.