Yes. All local variables (variables defined within a method) will be on their own Stack frame. So, they will be thread safe, provided a reference is not escaping the scope (method)
Note : If a local reference escapes the method (as an argument to another method) or a method works on some class level or instance level fields, then it is not thread-safe.
Does it mean that every thread gets a "copy" of the method even if it belongs to the same instance
No, there will be only one method. every thread shares the same method. But each Thread will have its own Stack Frame and local variables will be on that thread's Stack frame. Even if you use synchronize on local Objects, Escape Analysis proves that the JVM will optimize your code and remove all kinds of synchronization.
example :
public static void main(String[] args) {
Object lock = new Object();
synchronized (lock) {
System.out.println("hello");
}
}
will be effectively converted to :
public static void main(String[] args) {
Object lock = new Object(); // JVm might as well remove this line as unused Object or instantiate this on the stack
System.out.println("hello");
}