Let $D_X$ and $D_Y$ be the Chern classes of $L_X$ and $L_Y$. Assume $X$ and $Y$ have dimension at least one. Thus $pr_{1*}pr_1^*D_X=0$ and $pr_{2*}pr_2^*D_Y=0$. Applying the projection formula we obtain: $$pr_1^*D_X^{\dim X+1}=D_X^{\dim X}\cdot pr_{1*}pr_1^*D_X=0$$ $$pr_2^*D_Y^{\dim Y+1}=D_X^{dim Y}\cdot pr_{2*}pr_2^*D_Y=0$$ Now we compute the degree of $pr_1^*L_X\otimes pr_2^*L_Y$. It is same as the insersection number $(pr_1^*D_X+pr_2^*D_Y)^{\dim X+\dim Y}$. So, let us calculate: $$(pr_1^*D_X+pr_2^*D_Y)^{\dim X+\dim Y}=\sum_{i+j=\dim X+\dim Y} {\dim X+\dim Y \choose i}pr_1^*D_X^i\cdot pr_2^*D_Y^j$$. By the formulas in the beging all summands except ${\dim X+\dim Y\choose \dim X}pr_1^*D_X^{\dim X}\cdot pr_2^*D_Y^{\dim Y}$ vanish. Note, $pr_2^*D_Y^{\dim Y}=D_Y^{\dim Y}[X]$ and $pr_1^*D_X^{\dim X}=D_X^{\dim X}[Y]$, where $[X]$ and $[Y]$ are the classes of $X\times point$ and $point\times Y$ in the Chow ring of $X\times Y$. Easy to see that $[X]\cdot[Y]=1$. Finaly we find: $$(pr_1^*D_X\otimes pr_2^*D_Y)^{\dim X + \dim Y}={\dim X+\dim Y\choose \dim X}D_X^{\dim X}D_Y^{\dim Y}.$$ In the same way we compute degrees of $pr_1^*L_X$ and $pr_2^*L_Y$. $$pr_1^*D_X\cdot(pr_1^*D_X+pr_2^*D_Y)^{\dim X+\dim Y-1}={\dim X+ \dim Y -1 \choose \dim Y}D_X^{\dim X}D_Y^{\dim Y}$$ and $$pr_2^*D_Y\cdot(pr_1^*D_X+pr_2^*D_Y)^{\dim X+\dim Y-1}={\dim X+ \dim Y -1 \choose \dim X}D_X^{\dim X}D_Y^{\dim Y}$$

Let $D_X$ and $D_Y$ be the Chern classes of $L_X$ and $L_Y$. Assume $X$ and $Y$ have dimension at least one. Thus $pr_{1*}pr_1^*D_X=0$ and $pr_{2*}pr_2^*D_Y=0$. Applying the projection formula we obtain: $$pr_1^*D_X^{\dim X+1}=D_X^{\dim X}\cdot pr_{1*}pr_1^*D_X=0$$ $$pr_2^*D_Y^{\dim Y+1}=D_X^{dim Y}\cdot pr_{2*}pr_2^*D_Y=0$$ Now we compute the degree of $pr_1^*L_X\otimes pr_2^*L_Y$. It is same as the intersection number $(pr_1^*D_X+pr_2^*D_Y)^{\dim X+\dim Y}$. So, let us calculate: $$(pr_1^*D_X+pr_2^*D_Y)^{\dim X+\dim Y}=\sum_{i+j=\dim X+\dim Y} {\dim X+\dim Y \choose i}pr_1^*D_X^i\cdot pr_2^*D_Y^j$$. By the formulas in the begining all summands except ${\dim X+\dim Y\choose \dim X}pr_1^*D_X^{\dim X}\cdot pr_2^*D_Y^{\dim Y}$ vanish. Note, $pr_2^*D_Y^{\dim Y}=D_Y^{\dim Y}[X]$ and $pr_1^*D_X^{\dim X}=D_X^{\dim X}[Y]$, where $[X]$ and $[Y]$ are the classes of $X\times point$ and $point\times Y$ in the Chow ring of $X\times Y$. It is easy to see that $[X]\cdot[Y]=1$. Finaly we find: $$(pr_1^*D_X\otimes pr_2^*D_Y)^{\dim X + \dim Y}={\dim X+\dim Y\choose \dim X}D_X^{\dim X}D_Y^{\dim Y}.$$ In the same way we compute the degrees of $pr_1^*L_X$ and $pr_2^*L_Y$. $$pr_1^*D_X\cdot(pr_1^*D_X+pr_2^*D_Y)^{\dim X+\dim Y-1}={\dim X+ \dim Y -1 \choose \dim Y}D_X^{\dim X}D_Y^{\dim Y}$$ and $$pr_2^*D_Y\cdot(pr_1^*D_X+pr_2^*D_Y)^{\dim X+\dim Y-1}={\dim X+ \dim Y -1 \choose \dim X}D_X^{\dim X}D_Y^{\dim Y}$$

To compute the degrees of these bundles we don't need ampleness. Let $D_X$ and $D_Y$ be the Chern classes of $L_X$ and $L_Y$. Assume $X$ and $Y$ have dimension at least one. Thus $pr_{1*}pr_1^*D_X=0$ and $pr_{2*}pr_2^*D_Y=0$. Applying the projection formula we obtain: $$pr_1^*D_X^{dimX+1}=D_X^{dimX}\cdot pr_{1*}pr_1^*D_X=0$$ $$pr_2^*D_Y^{dimY+1}=D_X^{dimY}\cdot pr_{2*}pr_2^*D_Y=0$$ Thus, $\deg pr_1^*L_X=0$ and $\deg pr_2^*L_Y=0$. Now we compute the degree of $pr_1^*L_X\otimes pr_2^*L_Y$. It is same as the insersection number $(pr_1^*D_X+pr_2^*D_Y)^{dimX+dimY}$. So, let us calculate: $$(pr_1^*D_X+pr_2^*D_Y)^{dimX+dimY}=\sum_{i+j=dimX+dim_Y} {dimX+dimY \choose i}pr_1^*D_X^i\cdot pr_2^*D_Y^j$$. By the formulas in the beging all summands except ${dimX+dimY\choose dimX}pr_1^*D_X^{dimX}\cdot pr_2^*D_Y^{dimY}$ vanish. Note, $pr_2^*D_Y^{dimY}=D_Y^{dimY}[X]$ and $pr_1^*D_X^{dimX}=D_X^{dim X}[Y]$, where $[X]$ and $[Y]$ are the classes of $X\times point$ and $point\times Y$ in the Chow ring of $X\times Y$. Easy to see that $[X]\cdot[Y]=1$. Finaly we find: $$\deg pr_1^*L_X\otimes pr_2^*L_Y={dimX+dimY\choose dimX}\deg L_X\deg L_Y.$$

Let $D_X$ and $D_Y$ be the Chern classes of $L_X$ and $L_Y$. Assume $X$ and $Y$ have dimension at least one. Thus $pr_{1*}pr_1^*D_X=0$ and $pr_{2*}pr_2^*D_Y=0$. Applying the projection formula we obtain: $$pr_1^*D_X^{\dim X+1}=D_X^{\dim X}\cdot pr_{1*}pr_1^*D_X=0$$ $$pr_2^*D_Y^{\dim Y+1}=D_X^{dim Y}\cdot pr_{2*}pr_2^*D_Y=0$$ Now we compute the degree of $pr_1^*L_X\otimes pr_2^*L_Y$. It is same as the insersection number $(pr_1^*D_X+pr_2^*D_Y)^{\dim X+\dim Y}$. So, let us calculate: $$(pr_1^*D_X+pr_2^*D_Y)^{\dim X+\dim Y}=\sum_{i+j=\dim X+\dim Y} {\dim X+\dim Y \choose i}pr_1^*D_X^i\cdot pr_2^*D_Y^j$$. By the formulas in the beging all summands except ${\dim X+\dim Y\choose \dim X}pr_1^*D_X^{\dim X}\cdot pr_2^*D_Y^{\dim Y}$ vanish. Note, $pr_2^*D_Y^{\dim Y}=D_Y^{\dim Y}[X]$ and $pr_1^*D_X^{\dim X}=D_X^{\dim X}[Y]$, where $[X]$ and $[Y]$ are the classes of $X\times point$ and $point\times Y$ in the Chow ring of $X\times Y$. Easy to see that $[X]\cdot[Y]=1$. Finaly we find: $$(pr_1^*D_X\otimes pr_2^*D_Y)^{\dim X + \dim Y}={\dim X+\dim Y\choose \dim X}D_X^{\dim X}D_Y^{\dim Y}.$$ In the same way we compute degrees of $pr_1^*L_X$ and $pr_2^*L_Y$. $$pr_1^*D_X\cdot(pr_1^*D_X+pr_2^*D_Y)^{\dim X+\dim Y-1}={\dim X+ \dim Y -1 \choose \dim Y}D_X^{\dim X}D_Y^{\dim Y}$$ and $$pr_2^*D_Y\cdot(pr_1^*D_X+pr_2^*D_Y)^{\dim X+\dim Y-1}={\dim X+ \dim Y -1 \choose \dim X}D_X^{\dim X}D_Y^{\dim Y}$$