In all likelihood there will be a level of enjoyment or even fear in each case. However, it is only with a skilled facilitator and the ability to interact fully and freely with these amazing animals that self insights can be gained. It is then up to the individual where they go with these insights and the facilitator role is to help them process their insights in an un-judgemental and safe way.
Sadly, many organisations now believe that because they make available a few animals to pet occasionally they are providing some kind of animal therapy and can charge individuals accordingly or apply for funding on the back of this. They even make claims about social and psychological outcomes without any real research evidence to back this up.
At this point in time there are no moves to regulate this growing industry, and whilst I am a great believer in freely interacting with nature to feel a sense of connection, I would urge caution in thinking that because you have a horse or donkey you can deliver this kind of work without the appropriate knowledge and skills.