Is it permissible, in Islamic law, to conceal secrets and not discuss private matters with one's mother if she relays everything that happens in the house to others, especially when the relayed matters include very personal and embarrassing details? And is enduring this considered a part of showing filial piety?

Righteousness towards the mother and obedience to her in what is good is among the best acts of devotion and duties to Allah. The treatment for her disclosing household secrets should be gradual:

1. Advising her gently and kindly. 2. Concealing secrets from her or being evasive when she inquires about them. 3. Living in a separate house if you do not fear for her well-being and if you find someone to care for her.