In many states, police officers have a fair bit of leeway in deciding whether a person is intoxicated and, if so, whether the person should go to jail or a hospital, detoxification, or treatment center until they sober up. It often depends on your state's laws and whether your state treats public intoxication as a crime or a public health issue. A lot can also depend on whether treatment centers are available and have space to admit someone.
Generally, an officer can place intoxicated persons in "protective custody" if their level of impairment means they could be a danger to themselves or others. Protective custody refers to taking the person to a hospital, treatment center, detox facility, or even jail to sober up. Whether police can do that without the person's consent depends on the laws of the state and the person's capacity to make decisions.
Emergency custody. In some states, an officer can insist that an intoxicated person be admitted to a treatment center or hospital. But these situations are typically limited to emergency situations where the person no longer possesses the capacity to make their own decision.
Voluntary custody. If the person is coherent, usually the officer must get that person's consent before taking them to treatment. Unfortunately, not every jurisdiction has a detox center or similar facility and, even those that do, are often filled to capacity.
Sleep it off in jail. If no detox beds are available or the person refuses to consent to treatment, the officer may take the person into custody and make them sleep it off in jail. To do so, the officer must believe the intoxicated person is a danger to themself or others. While jail should be a last resort, it's often the only option in many places. If the person is intoxicated but not in any danger to themself or others, the officer should turn them over to a responsible adult or let them leave.
A person taken into protective custody is not being arrested or charged with a crime. However, don't assume the person can't still run afoul of the law. Despite there being no arrest, officers can still frisk the person if they suspect the person has a weapon. Whatever the officer finds, whether it's a weapon or drugs, can be used against the person. So while the person won't be arrested and charged with public intoxication, that doesn't mean officers can't arrest them for something else. The intoxicated person should also be careful as to what they say.
Yes, police can take a statement from an intoxicated person. And no, neither the person's intoxicated state nor the lack of a Miranda warning will automatically get that statement tossed out.
Police don't need to do much, if anything, to get an intoxicated person to volunteer information. And just because the person is intoxicated doesn't mean they don't know what they're doing or saying. It might be slurred or slightly incoherent but that doesn't make it inadmissible.
Also, despite what many think, police aren't required to give a Miranda warning every time they talk to someone. Police must read a person's Miranda rights only if two things happen (and it must be both): (1) The police detain or arrest the person, and (2) they ask the person questions that could lead to incriminating statements. If officers are taking you to a detox center, you haven't met criteria 1 because it's not an arrest. And if you're rambling on with no questions being asked, criteria 2 isn't met. This means whatever you say to officers—intoxicated or sober—may be admissible in a court of law.
Now, let's say you are arrested and police want to question you. The police give you a Miranda warning and ask if you understand your rights. If you indicate yes (even with a nod) and then keep on talking, you've given up your right to be silent. Being intoxicated won't save you. Courts have held that intoxication alone doesn't invalidate a confession or statement.
If you're stopped by a police officer while you are intoxicated, you should cooperate with the officer. (Don't resist arrest, as that can lead to serious charges and injuries.) If the officer says that you need to go to jail or to a hospital to sober up, it is worth asking (politely) whether you could call a friend to come get you or go home instead. However, in most situations, the police officer (not you) gets to decide the safest place for you while you are intoxicated.
If the officer asks you questions, try to stick to your personal information: name, address, date of birth, and reason for being there. Try not to offer up additional details in hopes that being super helpful will somehow be to your benefit. It's usually not.
If you are arrested for public intoxication or another crime involving alcohol, you should contact a criminal defense attorney. An experienced attorney can tell you the laws in your state and what to expect in court. With an attorney's help, you can hopefully achieve the best possible outcome in your case.