Both of them came up with the concept (semi-) independently; both were aware that the other was working on it and in fact they exchanged letters on the subject, but they approached it from different directions (Newton started with differentiation, Leibniz started with integration).
Going by their private papers it appears that Newton was further along earlier than Leibniz, but Leibniz published a full description much earlier than Newton did.
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No, Newton and Leibniz independently developed calculus.
Newton and Leibniz
Issac Newton and Leibniz
Isaac Newton and Gottfried Wilhelm Leibniz
Calculus was independently discovered by Sir Isaac Newton and Gottfried Wilhelm Leibniz in the late 17th century. Newton developed his own system of calculus called "fluxions" and "method of fluents," while Leibniz developed his system of calculus based on differentials and integrals. Both their work laid the foundation for modern calculus.