You can be charged with DUI on private property. Even in your driveway, you can often be charged with DUI. Public intoxication is a bit different and not usually a crime pursued in WA. Often, disorderly conduct or other charges are seen from public intoxication situations. The question is so open ended, it's rather difficult to answer. On private property, assuming you aren't disturbing the public peace, driving, or committing other law violations you are usually in a decent legal position being intoxicated. If you have more specific questions, I'd be happy to answer them further...
It doesn't make a lot of sense to me but the unfortunate truth is that you can be charged with a DUI on private property. The language in many statutes that relate to driving have eliminated the old language which once said "on a public highway" and have replaced it with "in the state of Washington." So unfortunately you can get charged with DUI on your own property.
I've never seen anybody charged with Public Intoxication on their own property but I guess it would depend on the facts and how much interaction you are having with the "public."
I agree with the other answers offered, but would add some context to the world of DUI and alcohol lawmaking and enforcement. First, DUI lawmaking is easy for our legislators - vote for anything that "makes our roads safer." Unfortunately, this pendulum swing has left some laws that tend to not make as much sense as they could. Under Washington law it is technically unlawful to mow your own law on a riding lawn mower (motor vehicle - I'm not kidding!) while exhibiting the effects of having consumed alcohol - this fits the definition of Driving Under the Influence. This is not to say this is a great idea, but it illustrates the point of how far the law can reach.
Second, public intoxication is often defined specifically by each municipality (city, town or county) and varies on the theme of 'don't be obnoxious in public.' Your yard is private property. Your sidewalk generally is not. Usually the worst that will happen to you is a talking to from an officer, but some places make certain debauchery a misdemeanor. If you're concerned about a barbeque, house party ,or the like it would be worth your time to review your local ordinances or even call your local Officer Friendly to get the official 'DOs' and 'DON'Ts.'