Here's the thing, I'm not sure what you mean by 'private time'. Do you mean sex? Do you mean masturbation? Do you mean lying in a bathrobe eating family bags of Skittles, cause that's my idea of private time. The fact that you have asked the question in this coy manner suggests it is not your son's embarrassment you need to consider but how to keep your own self consciousness at bay. Children are born pretty much immune to embarrassment, have you seen a toddler? They scratch, fart and pick their bogies with abandon. It is us grown ups that teach them to be ashamed of their thoughts and their bodies; sometimes this is a good thing but not always.
I have always known about sex or more accurately, I have no memory of learning about sex because there was no fanfare, no sitting down to have THE CONVERSATION. I believe sex and relationships talk should be littered into daily life in the same way one talks about the benefits of eating your greens - not necessarily the most exciting subject and in no way off limits. You want the tone to be honest, frank and really matter of fact; you don't want your child to think this is something to be fearful of overexcited by, the facts of life are indeed just a fact of life.
You might want to practise this, think about some of the more common questions your child may have and consider an honest but child centred answer you would feel comfortable delivering. Most of the awkwardness is generated by the shocked silence following a child's question about sex, be prepared and you can eliminate this potential pitfall.
A child is usually ready for answers when they start asking questions so in the meantime I think it's best to just show yourself to be an open, approachable parent. Have regular dialogues about friendships and feelings and make it clear that your child can ask you anything. If you are forced to confront an issue before the child is ready (say they walk in on you and a partner) simply stick to the facts necessary to avoid confusion. Again I would try and make it low pressure, you don't want the child to feel that they have done anything wrong. A friend once told me to approach awkward conversations from sideways, in the same way you would approach a horse (not that I ever have or would approach a horse but I can imagine). So activities that create connection without necessarily having to make eye contact - driving, cooking together, taking a walk - these can all be great opportunities to drop some knowledge on your kid, without them really noticing.
If you find that no matter how casual your approach your son shuts down and as he approaches his tweens relies on counsel from Google, buy a big notebook and tell your son that you want him to be able to come to you with any questions and if there is anything he wants to know to jot it down in the book and leave it in an agreed place. You can then take your time to deliver him a positive, empowering message that he can take in in his own time. He may never use this notebook, and that's okay, the important thing is that he knows that it's there.
photo credit: Beverly & Pack Sweet Nectar after a Light Sun Rain Shower, Pink Romantic Red Rose Petals & Landing Bumble Bee Guest Getting a Drink via photopin (license)