I agree with @Meg Coates' answer and would like to add my views.
The key here, is that she is talking at home.
This is not unusual and actually can be observed in people of all ages. Not everybody is comfortable speaking in social situations.
On the point of a three year old child. Children of this age are naturally suspicious of strangers and want to know that their primary care giver is close at hand, or if left alone, that the parent will be sure to return. Within this parameters of a, potentially, anxiety provoking experience, it is usual for some children to be reserved.
Children are prone to regressing when feeling stressed; or as the case may be here, shy.
My youngest child spoke her first word at about 15 months. "fluff". Our response was to comment and praise her as being so clever. She did not speak another word for over a year. She did not start speaking, until she could speak in full sentences and well. She took in all around her, studied and learnt.
It was then some years before any of my friends heard her speak. At this time, she would talk to her peers at preschool, but rarely adults, unless they were family.
Some thought there was something wrong with her. As she was my third child and I felt confident in my parenting abilities, I didn't fret, as I knew she was developing well, and that she had a very reserved nature. I respected that then, and still do. In fact, during this time, she became and still is very talkative at home.
As she grew older she gradually became more verbal and now speaks her mind freely in front of adults she knows well. She has the usual reservations of an eleven year old girl in front of adults she does not know well, which is also reflective of respect.
Basically, I have not forced her to talk to strangers, and supported her in setting her own pace when put into new situations, like starting school, starting a new sport. I trust in her nature, as a reserved person, that she is discerning and takes her time to assess people and situations before, wholly, participating in the activities. In a verbal and social sense.
This is an almost, endless, subject. I do know this. Many parents worry too much (that is our job) and I can say that what is "normal" or " healthy" development is broad. Contemporary psychology and studies, can place some children on the edges of the bell curve. It is important to remember this, it is just the propensity of human beings to strive for conformity. Fortunately human beings are made up of a great variety of personalities, and, usually, there is less to be concerned with, other than being overly-concerned.
nb. I have a child that had learning difficulties and he had speech therapy 5-6 days a week for a long time. The speech therapist and I would look at my daughter (a baby then) and would note how she was taking everything in. As for my son, when a child has difficulty learning, it is apparent, and a whole different thing. The parent is, usually, the best judge
side note by parent, I include and mean to acknowledge all primary caregivers.