Quiet and reserved: This is a direct opposite, emphasizing lack of volume and a tendency to keep thoughts private.
Subdued and introverted: This suggests a lack of assertiveness and a preference for quieter environments.
Modest and unassuming: This emphasizes humility and a lack of self-promotion.
Discreet and tactful: This highlights carefulness in speech and actions, avoiding drawing attention to oneself.
Diplomatic and conciliatory: This suggests a preference for compromise and avoiding confrontation.
Reserved and reticent: This implies a reluctance to share personal opinions or feelings.
Timid and shy: This conveys a lack of confidence and a fear of speaking up.
Meek and gentle: This emphasizes a peaceful and submissive nature.
The best antonym will depend on the context and what aspect of "loud and outspoken" you want to contrast.