Are poly1305 authenticators distinguishable from random data?
No, they are not distinguishable from random.

The Poly1305-AES authenticator is defined as:

$$
(((c_1 r^q + c_2 r^{q−1} + ... + c_q r^1 ) \bmod {(2^{130} - 5)}) + \operatorname{AES}_k (N)) \bmod 2^{128}
$$

Since it is the sum of an AES output and some other number modulo $2^{128}$, it is PRF if:


the AES output is PRF and
the two numbers are independent.


AES output looks like PRF as long as you see less than ~$2^{64}$ of them and $k$ and $r$ are required to be independent (since they are chosen from a uniform distribution), so the authenticators are indistinguishable from random at least as long as you see significantly fewer than $2^{64}$ of them.