A student who has completed a Master degree in Mathematical Sciences with specialization Logic has many possibilities. Usually, you have the same professional options as other graduates (with different specializations). If you are interested in research, you need not confine yourself to Mathematics departments: also Philosophy and Computer Science departments employ mathematicians and offer Ph.D. studentships for research in Logic. There are even possibilities in Linguistics!
Last but not least, a specialization in Logic may be interesting for students aiming for a career in teaching. I used to have a math teacher who told us, in 11th grade, about Cantor's diagonal method, Russell's paradox, and ordinal numbers, and I found this very exciting.