I beg your discretion on this matter.
I beg your discretion on this matter.

Hi. What does this sentence mean?
Thank you.

Question

I beg your discretion on this matter.
I beg your discretion on this matter.

Hi. What does this sentence mean?
Thank you.

Answer

Discretion = behave depends on situation or don't say anything offensive

Answer

Discretion, according to the dictionary, means: "behaving or speaking in such a way as to avoid causing offense or revealing private information"

So basically in your sentence it means - be careful  (1 don't talk about it to others 2 don't say anything offensive)
It might depend on the context. If it is sensitive because private - don't talk about it. 
if it is sensitive because the issue might be sensitive - related to - gender, race, religion, sex, etc - then means be careful in your choice of words regarding those topics.

Answer

This is very formal and means: "Please keep this quiet. Do not tell other people about this."

Answer

This means:
I am begging you to be discreet about this matter

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