As a Swedish national, I'm allowed to visit Moldova without a residence permit for 3 months in a 6-month period.
When going there I use my ID Card (accepted since September 2014), which cannot be stamped.
I've lately been asking myself: if I'm ever stopped for a police check, how can officers see that I've legally entered Moldova and not overstayed?
According to Moldova Border Police website:
Foreign citizens and stateless persons arriving in Moldova are recorded when crossing the state border and their identification data is introduced in the Population State Register.
So they will have the information in their system when you entered the country, and thus the officers have a way to check whether you entered Moldova legally and did not overstay.
If you're really concerned, you can carry the boarding pass with you, but I'd say there is no need really. Moldova is not a place with a large number of overstays, so if they suspect you did, they'd probably be really curious.
The only time it may be a concern is if you enter Moldova from Ukraine (via Transdnister region). In this case you will not go through the Moldova border control, and thus would have to register your data as following (same link):
Foreign citizens and stateless persons entering the Republic of Moldova through the Moldova-Ukraine border sector which is not under control of Moldovan authorities, are obliged to inform within 72 hours on their arrival at any population evidence and documentation unit (division) of the Ministry of Information Technologies and Communications. It is necessary to present the national identification document allowing entry in Moldova.
