I don't think this cold is contagious, so don't worry.
例えば、 I don't think this cold is contagious, so don't worry. 『この風邪は移らないと思うから心配しないで。』
他にも、 I'm sure this cold isn't contagious, so there's no need to worry. や This cold is not contagious, so you don't need to worry. という風にも言えます。
No need to worry, I don't think you'll get my cold.
この場合、次のような言い方ができます。
ーDon't worry, I don't think my cold is contagious.
「この風邪はうつりやすくないから心配しないで」
contagious で「うつりやすい・伝染性の」
ーNo need to worry, I don't think you'll get my cold.
「私の風邪はうつらないと思うから心配する必要はないよ」
to get one's cold で「人の風邪がうつる」
ご参考まで!