Where do we see centrifugal acceleration? A body in circular motion always possesses centripetal acceleration which is felt by a person sitting at the center of mass. It will not be felt by a person viewing the motion from the ground frame. Then where do we feel the centrifugal acceleration? We cannot be anywhere on the body itself because at every point we will perceive the motion to be circular by considering our point as the center.
 A: First of all, there is no such thing as "centrifugal acceleration".
I totally agree with what user Aaron Stevens mentioned in his answer.
I was previously mixing up two different topics, 
$1.$ The "pseudo" (inertial) centrifugal force which is used when the observations are made from a non inertial frame of reference like one where the relative motion of the frame and the body undergoing circular motion is $0$.
$2.$ The "reactive" centrifugal force which is present in any frame of reference because in any frame if the pivot applies a force (centripetal) on the rotating body transmitted via the interacting medium (string here), the body due to its inertia will apply an equal and opposite force to the pivot. 
Assuming you mean center of mass of the body, the person sitting on it will feel being pulled out while the centripetal force works to keep the body rotating.
From the usual experience, we can say if one sits in a car not sticking to it's walls which is turning to the left, they'll feel an apparent pull towards the right due to their inertia of direction. This can be seen from both the frames of reference (inertial and non inertial) but the non inertial observer uses the concept of pseudo (inertial) centrifugal force to explain it because according to him, the person moves to the right while the centripetal force is the only force to the left while the inertial observer argues that its just due to the person's inertia of motion in a straight line. 
Assuming the seat to be very smooth, when the person finally hits the right wall of the car as it turns, the car wall applies a centripetal force ($m\dfrac{v^2}{r}$) 
on the person's body and because of Newton's third law of motion, the person applies an equal and opposite force on the car ($-m\dfrac{v^2}{r}$) which is referred to as the "reactive" centrifugal force. Please note that this is true for any frame of reference.
Now I'd like to say one will feel pulled outwards undergoing a circular motion but that should be thought of due to inertia of the body to continue on a straight line while the centripetal force finally forces the body to move in a circle and thus giving rise to the "reactive" centrifugal force which is real and acting on the pivot or whatever constraint is forcing the motion.
The concept of centrifugal force was developed to get Newton's laws working in non inertial frames because Newton's laws are only valid for inertial frames.
However it should be kept in mind that in a non-inertial frame, the concept of pseudo forces can be used to get Newton's $1$st and $2$nd law working. But The $3$rd law requires interaction between bodies while the pseudo forces("inertial" centrifugal force here) are just a mathematical creation.
So the 3rd law can't be applied to fictitious forces.
A: Mathematically, the centrifugal force (as well as the Coriolis force) arises from working in a rotating reference frame. It is a fictitious, pseudo-, etc. force because it does not follow Newton's third law; it does not arise from any interactions, and it does not form an "action-reaction" force pair with another force acting on some other object.
Physically, if you are accelerating, you are going to "feel" these fictitious forces. i.e. going around a curve you feel pulled outwards. In an elevator accelerating upwards you feel slightly heavier for a moment (you feel an additional downward "force" acting on you). You can probably think of other examples.$^*$ But I would argue that your claim

We cannot be anywhere on the body itself because at every point we will perceive the motion to be circular by considering our point as the centre.

is false. You actually need to be on the accelerating body to feel these fictitious forces. For the example of circular motion, you feel pulled outwards by the centrifugal force and pulled inwards by the centripetal force. This results in no acceleration in your own reference frame, which should be the case (you are at rest relative to yourself).
While others in the comments seem to disagree with me, I will stand my ground. I agree that these non-inertial forces are not on the same level as actual forces that follow Newton's third law. But this does not mean they cannot be "felt" by someone who is accelerating. After all, by the Equivalence principle we cannot distinguish between accelerations and being in a gravitational field, and certainly we can feel gravity, right?

$^*$ I recently tried running across a spinning merry-go-round on a playground. I believe I felt a Coriolis effect as I did this, but it was hard to tell as the merry-go-round was not that big, and all of the railings to make the toy somewhat safer were in the way. It was still a fun experience though :)
A: Centrifugal force is a misnomer. The force that makes a body move on a curve is centripetal. The fact that you 'feel' a force is exactly analogous to feeling you are being forced back in your seat if you are in an accelerating car. There is no force pushing you back in your seat- the force is accelerating forwards.  
A: It's perhaps better, in terms of picturing it in your mind, to consider a merry-go-round, rather than a sphere or particle in an orbit.
Suppose you are standing on the merry-go-round as it's turning, near the edge. You have a ball in your hand. You let it go as if to drop it. To you, ignoring your brain's psychological tendency to tend to auto-reconcile it to that you're "really" just spinning, it flies away, toward and over the edge.
That acceleration is the centrifugal (not centripetal) acceleration. Centrifugal acceleration is only seen when you are actually in a rotating reference frame and that means either you are either on the rotating body indeed, or else you are rotating along with it by some other means. It's just that you won't see the rotating body as accelerating as by definition such means it is at rest with respect to you. You have to bring in another object and see how it behaves.
That said, you should not read too much into it. It's just a mathematical trick for describing the change of perspective resulting from such a reference frame shift in the framework of the usual (and perhaps less intuitive than it could be) formulation of Newtonian mechanics in terms of forces and acceleration. From the "ground frame", what is seen is the ball is cast off the edge by its tangential motion inherited from the rotation. Instead of such motion being due to the presence of some centrifugal "force", its rather due to the sudden absence of centripetal force.
And this is where and why I made that parenthetical regarding "perhaps less intuitive". In my mind, it would be better to forget about "forces" as an entity in themselves and instead to think about interactions - which makes much more sense when you get to more advanced physics, heck, even beyond just elementary mechanics, and to take stock of what is interacting with what and what the effects of such interactions are. In some cases, the effect of these interactions can be considered as what we'd otherwise call "exerting a force", but in others, it may be something different - such as heating something up, for example. Moreover, a "force", then, can only be said to properly "exist", then, when there is some interaction going on that has the effect that we would describe with that label.
In this picture, the only interaction is the centripetal one, between the rotating body and whatever is at the center of rotation, and the effect of this shows up in the ground frame as confining the object to move in circular motion, while in the rotating frame, its effect is to keep it from getting away: in a sense, just as the ground ("inertial") observer perceives everything to have a natural tendency to move in straight lines, the rotating frame observer "perceives" everything as having a natural tendency to want to go away from the center of rotation in spiral paths. In both cases, the centripetal interaction "exerts a force" in that it counteracts the natural motional tendency of the objects.
A: If you are sitting in any kind of vehicle which is moving along a curved path, then you feel the seat exert a centripetal force on you.  If you are swinging a mass around in a circle on the end of a rope, then the force which the rope exerts on you might be considered centrifugal.
