On incarceration rates, the US is the clear leader, with an incarceration rate of 716 people per 100,000 population, according to Wikipedia. Only the Seychelles (709) and St Kitts and Nevis (701) come anywhere close to this. There’s a big gap between these three and the rest of the world; next on the list is Cuba, with 510. Other large, wealthy democracies have figures like 148 (England and Wales), 146 (Scotland), 130 (Australia), 108 (Italy), 101 (France), 80 (Germany).
(These figures are for all prisons, not private prisons. I’m not sure whether your question 2 is directed to all prisoners, or just those in private prisons.)
As regards private prisons, there are different arrangements in different countries, and you need to define your terms carefully. A prison may be privately built, owned and operated. Or it may be privately financed and built, and then leased back to the state and publicly operated. Or it may be publicly owned, but the operation can be contracted out privately. There are different models of public/private partnership in different countries, and sometimes more than one model operating in a particular country. Some countries do not put many, or any, convicts in private prisons but do put, e.g., unconvicted immigration detainees in private prisons (or vice versa). These differences make makes inter-country comparisons difficult.
Still here’s a paper which attempts some comparisons: http://sentencingproject.org/doc/publications/inc_International%20Growth%20Trends%20in%20Prison%20Privatization.pdf
The table on page 2 suggests that the US is not the leader in the field, in terms of percentage of prisoners held privately; Australia is, followed by Scotland, England and Wales and New Zealand, with the US coming well down the field. On the other hand, very high US imprisonment rates plus a large population means that, in terms of raw numbers of prisoners, there are more privately-held prisoners in the US than in all other countries combined. Or, if we multiply the percentage of privately-held prisoners by national incarceration rates, we can see that 57 people out of every 100,000 in the US population are in private prisons, as compared with 21/100k in England and Wales, or 25/100k in Scotland and Australia.