Yes, to some extent we get the nature of the gods we worship. But it depends more upon the way we worship than whom we worship. So depending upon the method and mentality of worship there can be devotees of different nature. For example, there are devotees of Shiva who are satvic, but there are also devotees of Shiva like the naga babas who are generally tamasic. Hence, if one worships God with true devotion (without any personal desires), then instead of getting any of the three natures he gets beyond the three modes of material nature, viz. satva, raja and tama:
māṁ ca yo ’vyabhicāreṇa bhakti-yogena sevate
sa guṇān samatītyaitān brahma-bhūyāya kalpate [BG - 14.26]
One who engages in full devotional service, unfailing in all circumstances, at once transcends the modes of material nature and thus comes to the level of Brahman.
Moreover, it is the nature of the person that determines his likingness and preference regarding whom he worships:
yajante sāttvikā devān yakṣa-rakṣāṁsi rājasāḥ
pretān bhūta-gaṇāṁś cānye yajante tāmasā janāḥ [BG - 17.4]
Men in the mode of goodness worship the demigods; those in the mode of passion worship the demons; and those in the mode of ignorance worship ghosts and spirits.
So God is beyond the three modes. All these modes are the product of prakruti or material nature:
sattvaṃ rajastama iti prakṛterguṇāstair [SB - 1.2.23]
- Satva, raja, tama — these three are the qualities of prakruti or material nature.
sattvaṁ rajas tama iti guṇāḥ prakṛti-sambhavāḥ [BG - 14.5]
- Satva, raja, tama — these three originate from prakruti.
Hence, taking something as positive or negative is just our own way of perceiving things. And depending upon our own preference we attribute different deities and gods to them. From one point of view there is nothing good or bad because one is complementary to the other. Hence, it is a common known fact that Vishnu and Shiva are two interdependent deities. One cannot do without the other. Creation cannot last forever, one day it will get destroyed. This destruction is attributed to Lord Shiva and we say destruction is tamasic, but without it creation and its sustenance cannot happen again. So all are essential, it is the attitude we place towards them is what matters.