Who are you referring to when you say 'they'?
"They" in this context tends to mean either some form of government or their law enforcement arms.
OK thanks for clearing that up.
When you say that violent crime increases with areas of low gun ownership are you talking about on a country by country basis?
In the UK there is less violent crime than in the US yet less gun ownership?
You can look at it on a country by country basis, which actually doesn't bode well for the UK statistically. As a matter of fact, in 2013, gun crimes increased 34%, but overall violent crime rates are nearly 10x higher in the UK.
The problem with looking at violent crime stats is that the types of crime and reporting vary country to country. For instance, Stats Canada has different crimes and crime reporting than the US, so there may be a higher number of violent crimes listed by StatCan than the FBI uniform violent crime statistics. Actually, it is a higher number in Canada, nearly 2x, however a study showed that when trying to normalize the data between countries, which is actually somewhat difficult and introduces a little bit of subjectivity to the stats, Canada should technically have a slightly lower violent crime rate if compared apples to apples to the US.
Another problem with comparing stats between countries, it takes a relatively large population group and lumps them all together, so in particular with the US, you will find that by looking at things more locally, about 95% or so of the counties across the country have levels of violent crime in line with, or better than some of the most peaceful European nations. The problem is in the larger urban centers, like Washington DC, Chicago, Detroit, New Orleans, etc. They have a tendency to really skew the crime stats in the US as a whole because of an abnormally high rate of violence.
If you look closer at the US, you have to look at who is killing who. Mass shootings are an anomaly, though since Newtown, I know of at least 2 instances of attempted mass shootings in schools that were cut short by armed personnel on site, one of which was in Colorado. The rate of mass shootings in the US is relatively steady and it is difficult to find signal in the noise as to what the trends are.
Typical homicides in the US are generally gang related or drug related. Nearly half the homicides (firearm related or otherwise) are perpetuated by a demographic that only comprises of 5% of the US population (black people). Of that half of the homicides, about 75% to 80% (maybe more, I'd have to check the stats again) are black on black crime, many of which have past criminal behavior (this is actually relevant regardless of race).
Suicide wise, in Canada, you can see laws of substitution in effect. Many try to enact laws like "safe storage" laws to make it difficult for someone to use firearms as a means of suicide. From a lot of the research I've found, suicide by firearm is only about 3 to 5% more effective than suicide by hanging (the second most effective method). In Canada when we introduced safe storage laws, firearm related suicides were down, but suicides were not down. When you broke the numbers down, most of the suicides moved to hanging rather than firearms.
