I dislike the example of sharing personal information in the article. It's sometimes not only about your own comfort and boundaries, but also how much you want to put on another person.
I struggle sometimes with that, probably because I don't understand the 'put on another person' part. Would you (or anyone else reading this) care to give a recognizable example?
The "weight on one's 'shoulders'/'chest'" is a figure of speech from the psychosomatical relief one may feel when sharing troubles.
Inversely, the "burden" to respond, whether gravitas is implied or not, can be socially-exhausting to some, and not even conjured by others.
People with low self-esteem or self-worth / depressed / self-described "burdensome" people often assume the pity they may solicit costs the other party part of their own emotional well-being - they've burdened them further with expectations of a response or acknowledgement.