Your question regarding the Rest of the World hasn't really been answered. I can't give a comprehensive answer but I can give some indications for those countries in which I take an interest.
On the continent of Europe at the time the railways were developing the supply of private capital was not as great as was the case in Britain so many were built in the form of concessions granted by the Government or by royal decree or licence. To permit the concessionaire to make a profit very few competing lines were permitted by Government although in some cases the cities at the start and end of two routes might have been the same they both served very different areas of the country. This can be most clearly seen in France.
The Belgium railways were built to bind the country together. After the defeat of Napoleon in 1814 the Congress of Vienna created a new United Kingdom of the Netherlands to act as a buffer between France and Prussia but this split during the Belgian Revolution of 1830–1839 resulting in the creation of the three modern nations, Belgium, the Netherlands, and Luxembourg. The Belgium railways were planned by the state in the very early days of railways to link the areas with Brussels.
At the same time the railways in what is now Germany were planned and built by each state, Bavaria, Prussia, Hessen, Württemberg, Baden, Sachsen and so on, which made up the German Customs Union (
Zollverein). The routes in each State were generally centred on the respective capital city and the method of funding varied from place to place and line to line.
All this is a long way of saying that most railways in Europe had a great deal of Government involvement from the beginning and practically all of them are now Government owned. The Swiss private railways are often at least partially owned by the Cantons and other local authorities through which they run. Of course the railways in the USA were privately funded and constructed and I think that the Canadian Pacific route in Canada is also private.
Faced with the railways' declining market share the European Union wanted to bring market disciplines into railway operation. The first attempt was the Railway Directive 91/440/EC of 1991 which required that separate accounts for infrastructure and train operation must be kept and that trains run by other than the state administration should be permitted to use the tracks (Open Access). Subsequent Directives formed what is known as the First Railway Package and since then there have been the Second and the Third Railway Packages.
France went to great lengths to keep the SNCF as a monopoly but this is now cracking. In Germany the Federal Government changed the law to (mostly) comply with the EU Directives some years ago so that the individual States (
Länder) now commission local and regional services within their area. The successful bidders contract to operate a given number of train-km over a given route for a fixed term. The
Land may also specify the type of stock and service frequencies. In some cases the
Land acts as a ROSCO. All the usual suspects have won contracts, Abellio, National Express and so on. The
Länder claim that large savings have been made following competitive tendering but there are still questions about DB Netz, the infrastructure organisation, giving the DB passenger operations 'sweetheart' deals. In any event DBAG sees the operation of the S-Bahn networks as part of its core business and went to great legal lengths to delay the award of the Nürnberg S-Bahn to National Express so much so that NE had to back out. There is no Rail Regulator in Germany equivalent to the ORR so issues tend to get resolved in the courts.
There are many private freight haulage companies in Germany, some being subsidiaries of international companies and some private short lines.
In the last few years DBAG has lost its long-distance passenger monopoly, but the competition has had a rocky start and operates on only a couple of lines.
Do not think that everything in the foreign garden is wonderful. The reliability of long distance train operation in Germany has been woeful over the past few years (although it now seems to be improving slightly) and there are continual complaints about overcrowding.
The UK has the safest railway in Europe, see the
Eurostat statistics.