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In Hungary you may build a fire/have a barbecue while respecting the general rules of campfires (stay far from the forest area, clean the land of dry stuff before lighting a fire, put it out when you leave etc.). During the year the state officialities may issue "fire restriction periods" which are made public on websites and radio (but unfortunately only in Hungarian, yet you can follow them permanently here or here, together with general information about fires and regulations). This might occur when the weather is very hot or there was no rain for some time; in these periods of time (may last weeks) lighting a fire is strictly prohibited and severely punished with high fees.
In Romania the forestry may issue general rules (there must be at least 30 m distance from the closest forest, fire must be put out etc.) or special rules for certain areas (national parks or endangered areas - here usually fires are strictly prohibited and there are signs which tell you this when entering the area). Picnic and barbecue should be made in specially prepared places. Yet no rules are kept - people make fires wherever they like, and forestry doesn't really care about fining them for it. Details and fees can be found here (only in Romanian, unfortunately).