Phosphoserine phosphatase that mediates dephosphorylation of phosphoserine in the serine biosynthesis pathway. Also able to dephosphorylate other substrates such as phospho-L(or D)-threonine, with lower activity. Shows phosphoserine:homoserine phosphotransferase activity by transferring the phosphoryl group to homoserine using phosphoserine as the phosphoryl group donor. H2O + O-phospho-L-serine = L-serine + phosphate H2O + O-phospho-D-serine = D-serine + phosphate Binds 1 Mg(2+) ion per subunit. kcat is 13.4 min(-1) with O-phospho-L-serine as substrate. Amino-acid biosynthesis; L-serine biosynthesis; L-serine from 3-phospho-D-glycerate: step 3/3. Belongs to the thrH family. Its ability to complement a strain lacking the homoserine kinase thrB gene suggested that it acts as a homoserine kinase (PubMed:10220164). The ability to rescue a thrB mutant is explained by the phosphoserine:homoserine phosphotransferase activity in presence of phosphoserine, in a mechanism different from homoserine kinase, suggesting it is inappropriate to define it as a homoserine kinase (PubMed:14699121).