"It means that the act of delivering/telling the news will be difficult in itself."
Yes, it can mean that but it can also mean that after the telling the resulting responses may be sticky.
The OP "Breaking the news is going to be sticky." does put more emphasis on 'breaking' yet without any explanation it could still refer to the result. Compare: breaking the news to John is going to be sticky; breaking the news is going to be sticky for a lot of people.
And the use of 'sticky' is not how we would express this. We might use - difficult, awkward, explosive, inflationary, deflationary, demoralising, debilitating and a lot more whereas we would not use, say, uplifting or similar as an alternative. Sticky would be interpreted as unfavourable.