It depends.
For some
animals there are hints that they are sentient. Some possess the ability to speak or at least understand human speech. This ability is not like that of
our animal pet's. They understand us because of conditioning to words, sounds and other paraverbal communication. This is different for some animals in ME and requires a certain level of intelligence.
Some animals in ME are able to talk, like
wargs or
Ungoliant's descendants. However for those it could be argued that they have this ability only because of
Morgoth's corruption or because Ungoliant as an
evil spirit could.
For "regular" animals the most noticeable examples are
horses.
Shadowfax, as a
Mearas, was more intelligent than an average
horse. Mearas were brought to ME by
Orome and are said to be able to understand speech. Because they were originally from
Aman, they are a special case though!
Then there's Bill. According to
Gandalf he "learned" things in
Rivendell and was able to find his way back to
Bree. The horses in Rivendell might also be special cases since they live under the special circumstances of Rivendell, which is described by
Elrond as a "place of learning". But this tells us that regular horses are able to expand their intelligence in an environment that caters to their need to intellecutally evelove. This should require some sort of sentience.
Lastly in "The nature of
Middle-Earth" it is said that the animals that lived on Elenna before the
Edain arrived were very different from animals elsewhere. Even predatorian animals behaved more "friendly" and lived in peaceful cohabitation with the numenorians.
In contrast to this we can also see "sentience" as equivalent to "fea" and
Iluvatar only granted "fea" to very few animals. That would mean that apart from
the great eagles and
Huan there a no
known sentient animals .
In conclusion, it depends on the definition of "sentience", but some animals are shown to possess a form of intelligence one could associate with "sentience", while some are said to have been granted "fea", but there is no definitive proof all/most animals are sentient.