We consider stochastic differential equations on a Euclidean space driven by a Kunita-type semimartingale field satisfying a one-sided local Lipschitz condition. We address questions of local and global existence and uniqueness of solutions as well as existence of a local or global semiflow. Further, we will provide sufficient conditions for strong $p$-completeness, i.e. almost sure non-explosion for subsets of dimension $p$ under the local solution semiflow.