As usual. If you emphasise its existence, you use есть. If the core of your message is shifted elsewhere, есть is not used.
Since having a talking horse is rather surprising, the sentence in the title can omit есть in the same fashion as "He's got a gun!", i.e. when you stress the identity of what someone has rather than the fact that they own it.
Yesterday, Mosfet07 explained to me that with "У нас есть хорошие законы," adding есть suggests that some of the laws are good (but some are bad.) (as opposed to "У нас хорошие законы" which basically talks about all the laws)
So, could "У меня есть говорящая лошадь" also be used to mean that "of all the horses I own, I have one talking horse" ? (I realize that there is some missing context.)
With a stress on есть, you can. You really should listen to people talking to distinguish different types of meaning.
Long story short, in English you use "have" both to inform someone about possessing something and to describe traits:
- I have a car but do not have a house of my own.
- I have green eyes and fair hair. = My eyes are green and the colour of my hair is light.
We do not use есть in Russian in the second case. Now, one can assume that every country has laws. The difference is only what kind of laws these are. Saying «У нас хорошие законы» is essentially the same as saying "Our laws are good". An у-phrase at the beginning of a sentence is not always about possession.
As for domestic animals, it is not reasonable to expect everyne to have at least one horse. However, if it is established that a person owns a number of horses, they can indeed say the sentence in the title to tell you something about the talking horse they have—or say that most of their horses cannot speak.
An у-phrase signals relevance. We use it for diseases. We also tend to use it when indicating number of things or amount of something (thus shifting focus from the existence of something to its exact quantity).
A few examples from Tatoeba:
У нас закончились деньги. = We ran out of money.
У неё челюсть отвисла. = Her mouth fell open.
У нас полдюжины яиц. = We have half a dozen eggs.
У нас много друзей. = We have a lot of friends.
У неё простуда = She has a cold.
У его жены родились мальчики-близнецы. = His wife gave birth to twin boys.
У меня машина не заводится = My car would not start.
It is little wonder that such usage with statements of existence gets tricky for English speakers. I mean sentences like «У нас отличные лошади» where У нас can be interpreted as a "relevant" marker ("Our horses are top-notch") but you can also formally translate it as possession and sound OK ("We have excellent horses"). Actually, English also allows for a similar transformation in some expressions:
- I have my hands full. = My hands are full.
Алёша: Юлий? Я тебе знаю!
Я посмотрел "Три богатыря" когда я был маленький...