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The country could be policed just like any other country.

Police: Hey you, stop right there! Pedestrian: What is the problem, sir? Police: That was an illegal jaywalk you just performed. Do you have a license to perform illegal jaywalks? Pedestrian: Why yes, sir, I do. Here you go, have a look at this. Police: This license is expired! You are under arrest for "jaywalking with an invalid license." Put your hands behind your back. Pedestrian does so. Police cuffs and jails the heinous criminal.

Just like that.

A high percentage of gun ownership among the populace does not change anything that just happened. Most of the time, the firearm possessed by police does not even enter the equation and has no part in an event.

Potential problems?

Now let us examine the other extreme, where someone might think this could actually be a problem.

Assume a violent criminal possesses a firearm and that the law enforcement officer does not. This is the scenario in which trouble could arise. But think about that statement for a moment... "a violent criminal possesses a firearm." If this person is a violent criminal, then the fact that firearms are entirely legal is irrelevant. This person very well might have a firearm even if they were not legal.

So the exchange which could be problematic is not unique to a country where a pedestrian has a gun but police do not. In fact, this situation actually happens in reality. I recall an event, in London if I recall, a few years back in which two criminals were armed with a knife and a gun, and the local police, being unarmed, could do nothing but shout at them until the armed police unit arrived. The criminals were free to shoot at everyone until that time.

Potential benefits?

Now let's look at this from a different angle, about the possible benefit.

In the situation I mentioned in London, if a nearby pedestrian was armed with a gun, they could have helped the defenseless police instead of waiting and risking more lives. Or, if it is illegal in your hypothetical country for a pedestrian to do this, they could hand their gun over to the unarmed police to use in this life threatening situation, expecting it back after.

In fact, armed civilians have helped police in the past. There are instances of criminals attacking police and having them pinned down, where some nearby pedestrian has shot the criminal and saved the police.

Also, in a famous case where a pair of robbers wearing heavy full body armor were in a shootout with police, the police shots were not harming the criminals because of their heavy body armor. The police needed something better, and a local gun shop nearby handed over higher powered weapons for the police to use in the fight.

Also, in countries where firearms are illegal, the crime rates are generally not lower. Some people claim otherwise, pointing to gun-specific crimes being down, but the violent crime rates overall are generally not lower. In some such places, the crime rates are even higher, and interviewed criminals in prison have stated that they feel safer committing their crimes in those areas because they know the populace are not armed.

Conclusion

So how could they police the country? Well, they could allow civilians to take part in the policing. Or there could be a law that anyone who is armed and not in immediate danger must surrender their weapon to the police to use against a nearby threat (or it could just be a voluntary thing).

This could be a good thing as well, as it could help to reduce abuse by police. In most situations, the police are armed and those they are interacting with are not, so the police are able to easily bully people. This happens often. If you get too far out of line, they will draw their weapons. If they get too far out of line and you respond in your defense, they will draw their weapons. In your situation, people would be more free from the threat of police violence.

If you have a responsible populace, similar to Switzerland, this will likely be a more peaceful and safe place to live than what most of us are used to.

Some points to consider

Alexander asks,

"The problem here is not the legality of firearms, it's abundance of them. In day-to-day operations, police has to react to a number of incidents. Some of those incidents are involving guns. If number of those incidents are high (more than armed policemen can handle), should police dispatch unarmed officers to deal with them?"

That is a good question.

That depends on what you mean by "involving guns." If guns are being fired at people, then no, sending unarmed officers is just dumb. If by "involving guns" you mean "Police are responding to a non-violent crime, and the suspect just happens to have a pistol at their side or a rifle over their back," assuming you have a mature population, then yes, go ahead and send unarmed officers. The suspect is armed. So what? Police deal with armed suspects all the time without even knowing it in US states where open carry is illegal and everyone carries concealed; that has not been problematic.

Based on comments, I think there might be some confusion about my stance.

Obviously, it would be dumb to enter a dangerous situation while unprepared for it. I am often the first to say that we should be prepared for anything. I do not think it is wise to force the general police to be unarmed. The question states that there are unarmed police, so I am answering from that reality.

What I said is only meant to apply to people who have not been identified as violent. If you have any reason at all to believe that there could be violence, then yes it would be dumb to be unarmed. But the mere presence of a gun is not reason to believe that there could be violence. If the jaywalker in my answer's example had an AK-47 over their back but acted friendly, there is no reason to assume the worst. If they are known violent, or acting aggressive, or performed some worse crime that tends to lead to violence, I'd be concerned.

Similarly, if I was in Switzerland I would not feel in danger responding to a call where the suspect was armed but not deemed violent; but if I was in Mexico or certain Middle East areas or a known dangerous section of a major metropolitan area, I would feel in danger responding to any call, whether I knew a gun was present or not and I would not go unarmed.

Common sense is necessary. One of my points is just that the mere presence of a gun is not, in and of itself, cause for alarm. The appropriate response would be situational.