This one took more than a bit of research and emails (and google translate).
Taiwan oversees both, these strings of islands, are administered separately, Kinmen/Jīnmén 金門 by Kinmen County, Matsu/Mǎzǔ 馬祖 by Lienchaing County.
The answer is, yes, you can free camp on both Kinman and Matsu but that is a qualified 'yes,' as you shouldn't in any area with is military presence (so, it goes without saying: don't pitch your tent, or fire up the barbie there).
Wild camping is legal in Taiwan. Also known as free camping, you find a spot to set down your gear and stay, overnight or longer. Camping is hugely popular in Taiwan, and campsites, of all descriptions and prices, abound.
Taiwan is currently experiencing a “golden age” of camping, according to Morio Chen, honorary president of the Formosa Camping and Caravanning Club (FCCC). Every weekend, upwards of 300,000 people head for the hills and down to the beaches to set up their tents in order to hike, climb, river trace, surf, cycle, and visit hot springs.
For a quick, on-the-road reference, Wikitravel lists very affordable inns, guest houses, and homestays on both Kinmen and Matsu. Matsu National Scenic Area backpacking brochure is a handy take-along.
And a solid Wikitravel overview:
Informal camping is very easy and safe in Taiwan. Occasionally you can be scolded for choosing a bad spot, but threatening or violent behaviour is unheard of. It is generally okay to pitch a tent on the grounds of public schools, even in some larger towns. Just make sure to be gone before children arrive in the morning, and if there's a night guard posted, ask for permission first. Temple custodians may be willing to let you camp on their grounds or terraces as well, and you're unlikely to have any trouble camping next to an unstaffed shrine (certain ones may be considered undesirable because of evil spirits, but any objections will stem from concern for your well-being, not anger). Empty lots are of course fair game as well.
In national parks, there may be a rule against pitching tents outside of designated camping areas (which may or may not be present, and sometimes have a fee). However, it is considered normal to camp in parking areas or lawns anyway once the staff have gone home for the night. But do note that starting fires outside of designated fire pits is generally not considered appropriate in Taiwan (except on some beaches), and you should respect this. Taiwanese mountaineers cook and boil water using tiny portable gas stoves, which you can buy at outdoor sporting supply stores in major cities.
Edit: Also reference the WikiVoyage summaries for Matsu and Kinmen, with a thank you to @davidvc for the recommendation.
An email reply from the Matsu National Scenic Area Administration, Tourism Bureau, MOTC, shares this:
There are two camping sites running by local companies on Nangan island:
- Camping site at Mazu Religious Culture Park: providing bathrooms with shower facilities. Please contact Longfu Company( 0836-55661) for more information.
- Camping site at No.55 Hostel : providing power and bathrooms with shower facilities. Please contact No.55 Hostel via phone (0836-26426) or email (email@example.com) or searching on this website: http://bnb.matsu.idv.tw/No.55Hostel.
As for the campsite rules, please inquire managers from each site. However, please always be alert to snakes from April to September while camping in Matsu.
We wish you a pleasant journey in Matsu.