I'm not referring to someone you love that is very close to your heart, but people in general.
I'll hold my hands up and say I don't !
As an example, I would refer to the outpouring of orchestrated grief when Diana died - which I found to be pathetic and distasteful.
I can remember when my then g/f came into the bathroom, as I was shaving, to give me the news of her death I issued a single word reply - GOOD!
Imho, she was a slag and an embarrassment to the Royal Family and the UK.
That didn't, though, prevent the outpouring of what I call 'faux grief'.
The principle, though, is the same in everyday life BUT (to me) there is a big difference between politeness and OTT demonstrations of being seen/wishing to be seen to care.