both sides of the chest
Refers to the left and right sides of the chest area, not necessarily the breasts themselves.
事故で両胸を強く打った。
I hit both sides of my chest hard in the accident.
both breasts
Specifically refers to both breasts, often in medical or descriptive contexts.
授乳中は両胸が張ることがある。
Both breasts may become engorged during breastfeeding.
Compound of 両 (ryō, 'both') and 胸 (mune, 'chest'). The reading is a straightforward combination of the on'yomi of 両 and the kun'yomi of 胸.