Based on Google searches and interpreting forum posts on the Steam forums and the official Hitman forums:
- Codename 47 has a level-based checkpoint system: if you die, you need to restart the level.
- Silent Assassin has a restricted save system: you can only save a limited amount of times per level, based on the difficulty: 7 on normal, 3 on expert and 0 on professional.
- Contracts uses a similar system as as Silent Assassin. Note that there may be a bug on more modern systems where the game sometimes fails to properly save your game. A workaround that seems to work is to make your first save on normal difficulty. All saves after that work properly.
- Blood Money has the same save system as Silent Assassin and Contracts.
- Absolution has a checkpoint based system that autosaves based on your progress.
- Hitman (2016) allows you to save freely.