In the case of a statement as in your example then the 'or' implies that both parts are negative. Another example would be 'This house doesn't have a garage or a garden' which would mean that the house has neither a garage or a garden.
In the case of a question the 'or' usually means that either or both can be true. For example 'Does the Chinese lady speak English or German?' could be answered 'English', 'German', 'both' or 'neither'; this would be a question about the best way for Americans or Europeans to communicate with her. However the same question can have a different meaning in the case where there are alternative mother tongues as in "Does she speak Han or Cantonese" would be a question about which was her mother tongue.
For a property the question 'Does this house have a garden or a garage?' could be answered 'A garden', 'A garage', 'both' or 'neither' but the question 'Does this house have a tile roof or a slate roof' could be answered 'Tile', 'Slate' or 'Neither, it's thatched' as it's very unusual to have a house with a roof made from more than one material.