I am not following any particular tradition of Buddhism. How does anatta feel to a sentient being? How can it be tested or experimented? This is being asked because mere intellectualization is not enough obviously.
And what's next?
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The way Buddha formulated it, anatta was a surprising solution to the impossible problem - change in your attitude, not just an intellectual understanding - just like you said.
His argument went like this: dukkha happens because we try to rely on things which are unreliable -- therefore to achieve permanent Nirvana we need to try and find something which is absolutely reliable. Anything that depends on conditions is impermanent - therefore it is unreliable -- therefore it is dukkha. So we should not lean on (=try to establish our base in) anything conditioned (impermanent, unreliable). Let's see what it leaves us with. All physical objects, configurations, and situations are impermanent. So we can't establish our base in any of them. Our sense-experiences depend on the external objects, which are impermanent/unreliable - therefore our sense-experiences are unreliable too. So we can't establish our base in them. Our mind state depends on our sense-experiences, which are impermanent/unreliable - therefore the mindstate is unreliable too. So we can't establish our base in the mind state. This is the meaning of anatta, we decide that it would be stupid to establish our base in external world, in body, in sense-experiences, or in the mind - because they are all impermanent and therefore unreliable. So this leaves us with... nothing. There is nothing we can establish our base in. This is complete anatta, complete no-self / no-base. And then Buddha made discovery: if you don't establish your base anywhere, you are invulnerable! So on one hand it is a complete loss, there is nothing reliable so there is no Nirvana. On the other hand, if you really let go of everything, if nothing is yours, nothing is you, nothing is your base, nothing is your prize - then impermanence no longer matters, and there is no basis for dukkha to occur. This is called Unbinding.
So as you can see, anatta is not the kind of selflessness we get in physical sciences - they too tell us there is no soul, and lately the neuroscience tells us the feeling of single agent in our head making decisions and experiencing phenomena is an illusion. So everything is just a combination of matter/information fluctuating. But that's not enough.
What really matters is our personal attitude to things. As long as we establish our base anywhere, even in Buddhism - we get involved into conflicts, arguments, disappointments, and losses. Only if we have no base can we be free from conditions. But to be baseless requires tremendous courage and tremendous will power to let go (and to keep letting go, like jumping from the airplane). It really is very scary. And it kinda makes you very lonely, because you can't seriously relate with people who make their bases in all kinds of arbitrary things. On the other hand, since you don't dwell on any negative thoughts (you don't make them your base either) - you are never upset or frustrated.
It feels like you are superninja dancing between the raindrops. The raindrops represent various bases you could potentially attach to. Dancing between raindrops represents baselessness. When you get it, what are you going to do with this? What's next? At this point it becomes a funny question. Do I need a next base, is that's what I am asking about? What is my next base to go to?
What is my next address? What is Buddha's address? Buddha is not leaving - and Buddha is not arriving. Something like this. You asked how anatta feels to a sentient being.
But that's the very end of the path. Buddha said, there is no point in skipping stages - otherwise you will be like that cow - that has lost her previous grazing grounds before she has established her feet in the new grounds. First anatta should be practiced as baselessness in everyday life. Any time we feel frustration in daily life, that's our mind trying to establish a base. So we should learn to watch out for those and let go of the base, starting from the most coarse situations, that generate the most coarse emotions. And then you stalk yourself identifying more and more subtle bases, and let go of them, until you let go of the base of the Buddhist practice itself, and the base of "better", and the base of no base.
How does anatta feel to a sentient being? It feels like there is no self.
How can it be tested or experimented? Meditation brings it on, eventually. Shocking experiences can. Drugs can, but I do not recommend that and they do not bring understanding. Inquiry by someone who knows can startle or guide one in to a state of nonduality, or at least a breakthrough experience (often called "awakening" or "satori").
Even reading the right thing at the right time can. But as Andrei said, it is sometimes a realization after the fact - we realize that we have had the experience described. (This is sometimes called "gradual awakening".)
So, it is eminently do-able. "Try not to let your education get in the way of your learning." (Mark Twain)