both sides of the chest
Refers to the left and right sides of the chest area, not necessarily the breasts themselves.
I hit both sides of my chest hard in the accident.
both breasts
Specifically refers to both breasts, often in medical or descriptive contexts.
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.