According to this vintage ACNielsen report (download the pdf report), the global market-share for private brands was 17% in 2005 (* global meaning 38 countries and 80 categories), and had grown 6% that year.
Figure 1: Share & growth rates of private label by region (based on value sales) (source: ACNielsen, 2005)
Europe has the largest market-share with 23%, and Latin America the smallest, with 2%. Top countries included Switzerland, with 45%, Germany, with 30%, and the UK, with 28%. The largest Private Label growth happened, unsurprisingly, in emerging markets (11%) like Croatia (77%), Greece (24%), and Thailand (18%).
One big growth-contributor in Europe is the strong growth of hard discounters, such as Aldi or Lidl (both German chains), who are present in every European country and expanding rapidly. With Aldi, for instance, private labels make up 95% of sales.
In the emerging market like India, the impact is a lot lower at the moment due to the limited market share of modern trade, however, the trend is catching on in India as well.