* Some have no money for tickets to northern Europe

* Thousands of migrants are rescued at sea each week

* Many now waiting for German border controls to be lifted

By Steve Scherer and Emilio Parodi

ROME/MILAN, June 11 (Reuters) - Hundreds of migrants brought to Italy after being rescued at sea camped on streets near Rome's Tiburtina train station and gathered in Milan's main terminal on Thursday, making a brief stop on their journey to northern Europe.

At Tiburtina, migrants from Syria, Eritrea, Somalia and Ethiopia were among those resting in the shade under olive and fig trees. Children played in the supermarket nearby, and dozens napped on cardboard under an overpass.

Italy is struggling to handle the summertime surge in migrants, with thousands rescued at sea each week and more than 50,000 so far this year. Almost 2,000 have drowned.

Some northern Italian regions have refused to take in more migrants sent from crowded official reception centres down south, and now another crisis is brewing as local authorities struggle to deal with thousands heading north under their own steam.

"People say Italy is not good for migrants, so we want to leave," said Abdi Mohammed Adem, a 19-year-old Somali rescued 15 days ago by the Italian navy. His goal is to reach Germany or Britain, he said.

To send him to Europe, Adem's family sold its home and paid people smugglers $7,000, he said. But now he does not have enough cash to buy a ticket to go north.

"My family doesn't have any more money to give me," he said.

MONEY, BORDERS

The European Union has tripled funding for rescue missions in the Mediterranean after a shipwreck killed some 800 migrants in April, but it is still trying to find a way to cope with those who arrive, and to relieve the burden on southern countries like Italy and Greece.

Most migrants who reach Italy - 170,000 last year - want to reach northern European countries, which have stronger economies and generous welfare benefits. But many, like Adem, have run out of money.

Others are waiting until Germany suspends more stringent border controls still in place for a global summit held there.

Travellers can normally cross borders without checks in most of the European Union, but Germany imposed controls to prevent violent protests at a Group of Seven leaders held there at the weekend. The tighter controls will be lifted on Tuesday.

With many migrants exchanging tips and information on social networks, Milan's Stazione Centrale is another gathering point. An average of 800 arrive every night at the train station and are given beds at local shelters, said a city official.

Since October 2013, "64,000 have come to Milan on their own on their way to Germany and Sweden," Gabriella Polifroni, a spokeswoman for Milan's city council, told Reuters.

"There are always very many children among them," she said.

In the Milan station, meals are handed out and emergency medical care is provided. Unlike in Milan, no services at all are being provided to the migrants at the Tiburtina station, and some local residents are getting fed up.

"The institutions are nowhere to be seen here. It's pure chaos," said Michele Maggio, a local resident. "Either you don't allow them in the country or you set up the necessary structures to look after them." (Additional reporting by Gabriele Pileri and Cristiano Corvino; Editing by Tom Heneghan)