The (homeless) beggars are likely to congregate where they are most visible and likely to be successful at begging. Their visibility to tourists probably depends more on how willing the police is willing to crack down on them.
And your impressions of prevalence don't seem backed by data, at least before the financial crisis of 2008.
From "Homelessness in Europe and the United States:
A Comparison of Prevalence and Public Opinion (2007)":
Random samples of 250–435 adults were interviewed by telephone in five different
nations (N = 1,546): Belgium, Germany, Italy, the UK, and the United States. The
interview included questions on respondent attitudes, knowledge, and opinions
regarding homelessness; respondents’ own personal experiences with homelessness
and homeless people; and demographic characteristics of the respondents.
The highest rates for lifetime literal homelessness were found in the UK (7.7%)
and United States (6.2%), with the lowest rate in Germany (2.4%), and intermediate
rates in Italy (4.0%) and Belgium (3.4%). Less compassionate attitudes
toward the homeless were also found on many dimensions in the United States and
the UK. Possible explanations of these findings, drawn from various theoretical
perspectives, and policy implications are provided.
"Testing a Typology of Homelessness Across Welfare Regimes: Shelter Use in Denmark and the USA" (2015)
This article compares patterns of homeless shelter use in Denmark and the USA. Combining data from homeless shelters in Denmark with population registers, we find that the prevalence of shelter use is substantially lower in Denmark than in the USA. A cluster analysis of shelter stays identifies three types of users similar to findings from US research: the transitionally, episodically and chronically homeless. However, the transitionally homeless in Denmark have a higher tendency of suffering from mental illness and substance abuse than the transitionally homeless in the USA. The results support Stephens and Fitzpatrick' hypothesis that countries with more extensive welfare systems and lower levels of poverty have lower levels of homelessness, mainly amongst those with complex support needs, whereas in countries with less extensive welfare systems homelessness affects broader groups and is more widely associated with poverty and housing affordability problems.
It might have gotten worse in Greece.
I couldn't find stats on how many beggars are locals and how many intra-EU/EEA "migrants", but clearly the large difference in economic development between the Eastern EU (Romania, Bulgaria etc.) seem to play a role in the beggars you see in Paris or in Oslo.
At least according to one (2011) Swiss news article:
There is no national legislation on begging in Switzerland. It is left to the cantons and communes to deal with the problem.
The recent call for a ban in Lausanne follows similar moves in other parts of French-speaking Switzerland, such as Geneva, Vevey, Montreux, Renens and ten communes west of Lausanne, as well as cantons Fribourg and Neuchâtel. Other locations, such as Aigle, Yverdon and Pully, are also considering bans.
Geneva introduced a ban in February 2008 but groups of Roma beggars are still visible on its streets, despite several police round-ups and regular controls. It is estimated that the number of 200 previously present in front of stores and banks may have been cut by half.
Older begging bans are also in place in Basel, Zurich and Lucerne.
“Numbers clearly increased following the introduction of Schengen [25-country passport-free travel zone], but it’s still a small problem compared with Zurich and Bern,” said Klaus Mannhart, spokesman for the Basel City police.
Bern, meanwhile, has no ban against begging. But in June 2009 the police, along with Romanian and Bern city authorities launched a programme named “Agora” to crack down on organised gangs from eastern Europe targeting the city.
After almost 700 police checks - including 79 on children – totalling 2,000 working hours, Agora is considered a great success, said Bern aliens police chief Alexander Ott.
“We hardly have any more beggars; they say Bern and Switzerland – no more.”