How much should you drink around training?

Short answer
Start well hydrated, drink to thirst during training and replace what you lost afterwards. Guidelines advise against losing more than 2% of body weight through sweat. But do not drink so much that you gain weight during training.
Figures
- 2% · maximum body-weight loss through sweat
- 5–7 ml/kg · at least 4 h before training
- 1.25–1.5 l · per kg of weight lost, afterwards
Why 2%
From about 2% loss of body weight, endurance performance drops, especially in the heat. At 80 kg that is 1.6 kg. Lifting sessions rarely reach this.
Measure your sweat loss
Weigh yourself before and after a session, without wet clothes. Weight lost plus what you drank roughly equals your sweat loss; one kilogram is about one litre. That tells you how much you need in similar sessions.
Do not overdrink
Drinking more than you sweat during long exercise risks dangerously low blood sodium. One sign is weight gain during the session. For long sessions in the heat, drinks containing salt help.
Sources
Summaries by SetCtl, not medical advice.