When you get a lot of immigrants, of course there will be growth in sectors where lots of immigrants are employed. This is because there will be an increase it the supply of labor. If you can get more workers and/or cheaper workers, your sector will grow. The private sector is the sector most likely to include sectors that rely on immigrant labor.
As far as traded goods go, the demand for these will go up as population increases. There is nothing special about immigration that will make this sector grow -- any increase in population would do the same.