I'm not sure I should admit this in public, but although I am aware of the precise formulations using first order logic and beyond (mentioned in the above answers), I tend not to use them. Rather I view the Lefschetz principle as more of a philosphical principle of what ought to be possible in general, and do the necessary verifications as and when I need them (but perhaps only implicitly). I suspect this attitude is pretty common among many algebraic geometers.
To give an example, for many years the only known proofs* of the Kodaira vanishing theorem were analytic. But since coherent cohomology behaves well under field extensions, Kodaira vanishing is valid over arbitrary (not necessarily algebraically closed) fields of characteristic $0$. On the other hand, for certain kinds of arguments, one needs a big enough field to carry out the argument. This typically happens when one is forced to remove a countable union of exceptional sets. Curiously, the Noether-Lefschetz theorem is one such case. Here the Lefschetz principle in the most naive sense won't work.
*(Added Footnote.) There is now an algebraic proof due to Deligne and Illusie, which involves reduction to positive characteristic. This is yet another kind of transfer.