His father Vlado is a former president of the Croatian Water Polo Federation, while his brother Dario is a former Croatian water polo player. He is married Antonia Kojan.[citation needed]
On 3 February 2000, Kobeščak was sentenced in Municipal Court in Croatia to a one year prison sentence. He was convicted of a serious traffic accident that happened on 6 November 1993, in which his passenger Marijana Matošin was killed.[citation needed]