The trick lies in appearing to be a hands off parent.
From a safe distance, I let my baby daughter make her own choices and actions —within the scope of her ability. I would play with her at frequent intervals, and then sit back. As she grew older our play intervals grew shorter, though supervision remained the same.
From the time she could sit up, I positioned toys, baby bottle, cookie snacks, etc., where she would have to deliberately turn or reach to use whichever she wanted. If she threw one away, I did not retrieve it. She would get it back after a nap, a meal, or other change in her day. As she reached the crawling stage, I positioned toys (according to her mobility) around the room. She had to crawl to them to use them.
This popular YouTube video shows how thoroughly a child can entertain his or her self, despite a minimum of parental interaction (edited out):
I did not want my daughter to rely on my being at her disposal. I wanted her to choose, entirely on her own, what she was going to do next. Doting parents stifle a child’s mind. There was a side benefit in her entertaining herself. I could get other things done, nearby, and could reduce my time at baby play, which I found unbearably boring (undiagnosed sleep apnea had me falling asleep on the floor beside her).
She quickly learned not to throw away toys she might want later, and would choose what she wanted to do, rather than cry for me to entertain her. If she threw away a toy, she had her own motivation to retrieve it.
The consequences were almost miraculous. As a toddler, if she was being bothered by her older sister, she would take the toys she wanted to a different room, without a fuss. If she wanted something, she would ask without whining. If you asked her to wait, she would do something else.
Her independence extended to how she related to friends. I promoted sharing & helping as something one does with friends (people she valued) ─but some personal & important things are not for just anybody. To this end, when friends were coming over, we would choose what toys should not be shared, and put them away. The combination of independent play & chosen sharing worked well when she began school.
At ~3.0 yrs of age she entered Montessori Casa-1. She immediately took to the idea of working on her own ‘manipulatives’. If other children bothered her she would ask them to leave her alone, or ask them why they were not playing with their own manipulative. It was natural to her not to bother them.
If a classmate did not see how to use a manipulative, she would help. Her motivation to help was not a dependence on friendship or obedience to required niceness, it was simple decency. Her attitude set an example that helped the other children focus. By grade three she would try to get her homework done while in the car driving home, because it was hers to do. Then she could play with friends, once home.
Ten years later, her daily behavior reflects that independence of thought and action.
Have you had a similar success using a different method?
[Update: added video ─very relevant─ & minor edits that required.]